Computation of geometric partial differential equations and mean curvature flow

Transcription

1 Acta Numerica (2005), pp c Cambridge University Press, 2005 DOI: /S Printed in te United Kingdom Computation of geometric partial differential equations and mean curvature flow Klaus Deckelnick Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, D Magdeburg, Germany [email protected] Gerard Dziuk Abteilung für Angewandte Matematik, Albert-Ludwigs-Universität Freiburg i. Br., Hermann-Herder-Straße 10, D Freiburg i.br., Germany [email protected] Carles M. Elliott Department of Matematics, University of Sussex, Mantell Building, Falmer, Brigton, BN1 9RF, UK [email protected] Tis review concerns te computation of curvature-dependent interface motion governed by geometric partial differential equations. Te canonical problem of mean curvature flow is tat of finding a surface wic evolves so tat, at every point on te surface, te normal velocity is given by te mean curvature. In recent years te interest in geometric PDEs involving curvature as burgeoned. Example of applications are, amongst oters, te motion of grain boundaries in alloys, pase transitions and image processing. Te metods of analysis, discretization and numerical analysis depend on ow te surface is represented. Te simplest approac is wen te surface is a grap over a base domain. Tis is an example of a sarp interface approac wic, in te general parametric approac, involves seeking a parametrization of te surface over a base surface, suc as a spere. On te oter and an interface can be represented implicitly as a level surface of a function, and tis idea gives rise to te so-called level set metod. Anoter implicit approac is te pase field metod, wic approximates te interface by a zero level set of a

2 2 K. Deckelnick, G. Dziuk and C. M. Elliott pase field satisfying a PDE depending on a new parameter. Eac approac as its own advantages and disadvantages. In te article we describe te matematical formulations of tese approaces and teir discretizations. Algoritms are set out for eac approac, convergence results are given and are supported by computational results and numerous grapical figures. Besides mean curvature flow, te topics of anisotropy and te iger order geometric PDEs for Willmore flow and surface diffusion are covered. CONTENTS 1 Introduction 2 2 Some geometric analysis 12 3 Definition and elementary properties of mean curvature flow 17 4 Parametric mean curvature flow 19 5 Mean curvature flow of graps 28 6 Mean curvature flow of level sets 36 7 Pase field approac to mean curvature flow 43 8 Anisotropic mean curvature flow 51 9 Fourt order flows 70 Appendix 85 References Introduction A geometric evolution equation defines te motion of a ypersurface by prescribing te normal velocity of te surface in terms of geometric quantities. As well as being of striking matematical interest, geometric evolution problems occur in a wide variety of scientific and tecnological applications. A traditional source of problems is materials science, were te understanding of te strengt and properties of materials requires te matematical modelling of te morpology of microstructure. Evolving surfaces migt be grain boundaries, wic separate differing orientations of te same crystalline pase, or solid liquid interfaces exibiting dendritic structures in under-cooled solidification. On te oter and newer applications are associated wit image processing. For example, in order to identify a dark sape in a ligt background in a two-dimensional image a so-called snake contour is evolved so tat it wraps around te sape. In tis article we survey numerical metods for te evolution of surfaces wose normal velocity is strongly dependent on te mean curvature of te surface. Te objective is to find a family {Γ(t)} t [0,T ] of closed compact and orientable ypersurfaces in R n+1 wose evolution is defined by specifying te velocity V of in te normal direction ν. An example of a general geometric

3 Computation of geometric PDEs and mean curvature flow 3 evolution equation is V = f(x, ν, H) on Γ(t), (1.1) were f depends on te application and te x dependence migt arise from evaluating on te surface Γ(t) field variables wic satisfy teir own system of nonlinear partial differential equations in R n+1 away from te surface. It is important to note tat, in order to specify te evolution of te surface, it is sufficient to define te normal velocity. Te prototype problem is motion by mean curvature, forwic V = H on Γ(t), (1.2) were H is te sum of te n principal curvatures of Γ(t). We call H te mean curvature rater tan te aritmetic mean of te principal curvatures. Our sign convention is tat H is positive for speres, wit ν being te outward normal. It is well known tat, starting from an initial surface Γ 0, tis equation is a gradient flow for te area functional, E(Γ) = 1dA. (1.3) In applications te area functional is an interfacial energy wit a constant energy density 1. Equation (1.2) may be viewed as an analogue for surfaces of te parabolic eat equation u t u =0. On te oter and, anoter geometric equation is V = Γ(t) H on Γ(t), (1.4) were Γ(t) is te Laplace Beltrami or surface Laplacian operator on Γ(t). Tis can be viewed as an analogue of te spatially fourt order parabolic equation u t + 2 u = Approaces In order to solve a surface evolution equation analytically or numerically, we need a description of Γ(t). Eac coice of description leads to a particular nonlinear partial differential equation defining te evolution. Tus te computational metod depends strongly on te way we coose to describe te surface. For tis article we sall focus on four possible approaces. Parametric approac. Te ypersurfaces Γ(t) aregivenas Γ(t) =X(,t)(M), were M is a suitable reference manifold (fixing te topological type of Γ(t)) and X : M [0,T) R n+1 as to be determined. Here X(p, t), for p M, is Γ

4 4 K. Deckelnick, G. Dziuk and C. M. Elliott Figure 1.1. A dumbbell-saped two-dimensional surface parametrized over te unit spere. Figure 1.2. A lemniscate, parametrized over te unit circle. te position vector at time t of a point on Γ(t). If we are interested in closed curves in te plane ten M can be te unit circle S 1, wereas if Γ(t) isa two-dimensional surface ten M could be te unit spere S 2 (see Figures 1.1 and 1.2). Geometrical quantities are easily expressed as derivatives of te parametrization so tat evolution laws suc as (1.2) may be translated into nonlinear parabolic systems of PDEs for te vector X. Wit tis approac tere is no notion of te surface being te boundary of an open set and aving an inside and outside, so self-intersection is perfectly natural for smoot parametrizations and is not necessarily associated wit singularities. For example in te plane a figure of eigt curve can be smootly mapped onto te unit circle one to one (Figure 1.2). At te crossing point te curve as two smootly evaluated normals and curvatures wic depend on te parametrization. A parametrized curve evolving by mean curvature can evolve smootly from tis configuration. Graps. We assume tat Γ(t) can be written in te form Γ(t) ={(x, u(x, t)) x Ω}, were Ω R n and te eigt function u : Ω [0,T) R astobe found. We sall see tat te law (1.2) leads to a nonlinear parabolic PDE for u. Clearly, te assumption tat Γ(t) is a grap is rater restrictive; owever, tecniques developed for tis case ave turned out to be very useful in understanding more general situations. Since te eigt is a smoot function we can view Γ(t) as dividing Ω R into two sets, namely te regions above and below te grap.

5 Computation of geometric PDEs and mean curvature flow 5 Figure 1.3. Level lines of a level set function (rigt) for te figure of eigt curve (left). Figure 1.4. Grap of a level set function for te figure of eigt curve, cut at te zero level. Negative part left and positive part (grapically enlarged) rigt. Level set metod. We look for Γ(t) as te zero level set of an auxiliary function u : R n+1 [0, ) R, tat is, Γ(t) ={x R n+1 u(x, t) =0}. Te law (1.2) now translates into a nonlinear, degenerate and singular PDE for u. Clearly intrinsic to tis approac is te notion of Γ(t) being a dividing surface between te two regions were te level set function is positive and negative. Tus we ave te notion of inside and outside. In order to describe a figure of eigt by a level set function it is necessary to ave te level set function positive and negative, as sown in Figures 1.3 and 1.4. Pase field approac. Te pase field approac is based on an approximation of te sarp interface by a diffuse interface Γ ɛ (t) ={x R n+1 1+Cɛ u ɛ (x, t) 1 Cɛ} of widt O(ɛ), across wic te pase field function u ɛ as a transition from approximately one bulk negative value 1 to approximately a second positive bulk value +1. Te zero level set of te pase field function approximates te surface. Just as in te level set metod tere is te notion

6 6 K. Deckelnick, G. Dziuk and C. M. Elliott of a material interface separating an inside and outside and in te basic implementation interface self-intersection and topological cange are andled automatically. Te bulk values of te pase field function correspond to te minima of a omogeneous energy function wit two equal double wells. Interfacial energy is assigned to te diffuse interface via te gradient of te pase field function. For motion by mean curvature te evolution is defined as a semilinear parabolic equation of reaction diffusion or Ginzburg Landau type. Frequently in applications matematical models are derived wic, from te beginning, involve diffuse interfaces and pase field functions. Comments Conceptually te grap formulation is te simplest and most efficient. It involves solving a scalar nonlinear parabolic equation in n space dimensions and directly computes te surface. However, tere are many circumstances were te surface is not a grap. Furtermore, even if te initial surface is a grap it is possible tat over te course of te evolution tat property migt be lost, despite te surface evolving smootly. Tis would lead to gradient blow-up of te solution of te grap equation. Tere is te possibility tat te solution of a numerical discretization exists globally and appears to be stable even toug tere is no solution to te continuous equation. Te parametric approac is also direct. It is conceptually more advanced tan te grap approac and one as to solve in n space dimensions a system of n + 1 parabolic equations. If te surface is a grap ten te parametric approac is less efficient tan solving for te eigt of te surface. On te oter and it is more widely applicable. In te case of a closed curve one can use periodic boundary conditions on te unit interval in order to solve over te circle. A closed two-dimensional surface can be approximated by a polyedral surface. A parametrized surface does not see an inside or outside. From te point of view of differential geometry tis may not be an issue. However, wen te surface separates two pases, or two materials, or two colours, tere are significant issues. For example, consider using two colours in Figure 1.2 in order to define te curve as te interface between te coloured regions. Black may be used to colour te inside of bot loops and wite to colour te te rest of te plane, but if black is used inside just one loop ten te oter loop is lost. Tus, in order to use te parametric approac wit tis initial condition, one eiter tinks of a parametrization wic traverses te curve witout a crossing, but wit a single self-intersection, or regards tem as being two separate closed curves wic touc at one point. Tese coices lead to differing evolutions for mean curvature flow. Contrary to te parametric approac, te level set metod as te capability of tracking topological canges (like pincing-off or merging) of Γ(t) in an automatic way. In te basic implementation of te metod topological

7 Computation of geometric PDEs and mean curvature flow 7 cange is noting special and is observed in post-processing te computational output. Tis is because, in principle, zero level sets of continuous functions can exibit tese features. However, from te matematical point of view tere are issues of existence of solutions of te degenerate partial differential equations tat te level set approac generates. In te case of motion by mean curvature tere is te notion of a viscosity solution wic yields a unique evolution from any continuous function. Te example of te lemniscate discussed in te context of te parametric approac introduces a new idea in te level set approac of fattening of te interface. Te level set for tis example develops an interior wose boundary yields bot of te described solutions. Self-intersection, merger and pinc-off can all be simulated by tis approac. Tis advantage, owever, needs to be offset against te fact tat te problem now becomes (n + 1)-dimensional in space. Te pase field approac can also andle topological cange, self-intersection, merger and pinc-off witout doing anyting special. It is te one approac wic in its conception involves an approximation. Te fact tat it involves a new parameter ɛ is bot an advantage and a disadvantage. Te parabolic equations are in principle easy to solve but possess a certain computational stiffness due to te tickness of te diffuse interface. However, in many applications pase field models arise naturally and te ɛ parameter allows us to resolve singularities in a way wic may be viewed as being pysically motivated. From bot te matematical and pysical points of view it is widely applicable in a rational way, wereas te use of te level set metod is frequently ad oc. In general, te coice of one or te oter approac will depend on weter one expects topological canges in te flow Applications In wat follows we list some problems in wic a law of te form (1.1) or generalizations of it arise. Grain boundary motion Grain boundaries in alloys are interfaces wic separate bulk crystalline regions of te same pase but wit differing orientations. Associated wit te grain boundary is a surface energy wic gives rise to a termodynamic restoring force. For a constant surface energy density tis is simply te surface tension force proportional to te mean curvature and te resulting evolution law is just (1.2). Frequently tere is also a driving force causing motion of te grain boundary. Surface growt Te growt of tin films on substrates is tecnologically important. For example, epitaxy is a metod for growing single crystals by te deposition

8 8 K. Deckelnick, G. Dziuk and C. M. Elliott of atoms and molecules on to a growing film surface. Tere are numerous pysical mecanisms operating at differing time and lengt scales wic affect te growt process. A simple model would ave a driving force representing te deposition flux of atoms onto te surface wic migt be in te normal direction or in a fixed vertical direction parallel to a beam of arriving atoms. Image processing One of te most important problems in image processing is to automatically detect contours of objects. We essentially follow te exposition of Aubert and Kornprobst (2002). Suppose tat M R n+1 (n = 1 or 2) is a given object and let I(x) =χ Ω\M (x) be te caracteristic function of Ω \ M. Te function 1 g(x) = 1+ I σ (x) 2, were I σ is a mollification of I, will be small near te contour of M. It is terefore natural to look for minimizers of te functional J(Γ) = g(x)da Γ were Γ is a curve in R 2 or a surface in R 3. Te corresponding L 2 -gradient flow leads to te following evolution law: find curves/surfaces (moving snakes ) Γ(t) suc tat V = (gν)= gh g ν on Γ(t). Here, t plays te role of an artificial time; clearly tis law fits into te framework (1.1). Stefan problem for undercooled solidification Consider a container Ω R n+1 (n = 1 or 2) filled wit an undercooled liquid. Solidification of te liquid follows te nucleation of initial solid seed wit caracteristic diameter larger tan te critical radius. Te seed will ten grow into te liquid. A matematical model for tis situation is te Stefan problem wit kinetic undercooling, in wic te solid liquid interface is described by a curve/surface Γ(t) and as to be determined togeter wit te temperature distribution. Here te interior of Γ(t) is te solid region Ω S (t) and te exterior is te liquid region Ω L (t). Using a suitable non-dimensionalization te problem ten reads: for a given initial pase boundary Γ 0 and initial temperature distribution Θ 0 =Θ 0 (x) (x Ω), find te non-dimensional temperature Θ = Θ(x, t) and te pase boundary Γ(t) (t >0), suc tat te eat equation is satisfied in te bulk, tat is, Θ t Θ = 0 in Ω \ Γ(t),

9 Computation of geometric PDEs and mean curvature flow 9 togeter wit te initial value Θ(, 0) = Θ 0 in Ω. On te moving boundary te following two conditions are satisfied: V = 1 [ ] Θ on Γ(t), (1.5) ε l ν Θ+ε V β(ν)v + σh γ =0 onγ(t). (1.6) Here, [ Θ/ ν] denotes te jump in te normal derivative of te temperature field across te interface and ε l is te constant measuring te latent eat of solidification. Equation (1.6) is te Gibbs Tomson law; ε V,σ are non-dimensional positive constants measuring te strengt of te kinetic undercooling and surface tension wic depress te temperature on te solid liquid interface from te scaled equilibrium zero melting temperature. Furtermore, H γ is an anisotropic mean curvature associated wit a surface energy density, γ(ν), depending on te orientation of te normal. Tere may also be anisotropy, β(ν), in te kinetic undercooling. Note tat (1.6) can be rewritten as ε V σ β(ν)v = H γ 1 σ Θ on Γ(t). If we consider Θ as being given, tis equation again fits into our general framework (1.1) provided we allow for a coefficient in front of V and a generalized notion of mean curvature. Figure 1.5 from Scmidt (1996) sows a simulation in wic te free boundary was described by te parametric approac resulting in a sarp interface model. One can see te free boundary forming a dendrite. For Figure 1.5. Evolution of a dendrite wit sixfold anisotropy. Time-steps of te free boundary (left) and adapted grid for te temperature at one time-step (rigt).

10 10 K. Deckelnick, G. Dziuk and C. M. Elliott results concerning tree-dimensional dendrites and more information about te algoritm we refer to Scmidt (1996). Figure 1.6 from Fried (1999) illustrates a possible effect of using a level set metod for te free boundary in tis problem. Dendrites may seem to merge. But if a smaller time-step is used te dendrites stay apart. For more information about a level set algoritm for dendritic growt we refer to Fried (1999, 2004). Surface diffusion and Willmore flow Te following laws do not fit into (1.1), but we list tem as examples of important geometric evolution equations in wic te normal velocity depends on iger derivatives of mean curvature. Te surface diffusion equation V = Γ H (1.7) models te diffusion of mass witin te bounding surface of a solid body. At te atomistic level atoms on te surface move along te surface owing to a driving force consisting of a cemical potential difference. For a surface wit constant surface energy density te appropriate cemical potential in tis setting is te mean curvature H. Tis leads to te flux law ρv = div Γ j, were ρ is te mass density and j is te mass flux in te surface, wit te constitutive flux law (Herring 1951, Mullins 1957) j = D Γ H. Here, D is te diffusion constant. From tese equations we obtain te law (1.7) after an appropriate non-dimensionalization. In order to model te Figure 1.6. A possible effect of te use of a level set metod. Growing dendrites: merging (left) for large time-step size and staying apart (rigt) for smaller time-step size.

11 Computation of geometric PDEs and mean curvature flow 11 underlying structure of te solid body bounded by Γ, anisotropic surface diffusion is important, tat is, V = Γ H γ, (1.8) wit H γ denoting te anisotropic mean curvature of te surface Γ as it is introduced in (8.15). A similar evolution law is Willmore flow, V = Γ H + H Γ ν H3 on Γ(t), (1.9) wic arises as te L 2 -gradient flow for te classical bending energy E(Γ) = 1 2 Γ H2 da. Apart from applications in mecanics and membrane pysics tis flow as recently been used for surface restoration and inpainting Outline of article Tis article is organized as follows. In Section 2 we present some useful geometric analysis, in particular te notion of mean curvature. Te basic mean curvature flow is defined in Section 3 and some elementary properties are described. Te next four sections consider in turn basic approaces for numerical approximation. In Section 4 we consider te parametric approac. We start wit te classical curve sortening flow and present a semidiscrete numerical sceme as well as error estimates. Next, we sow ow to apply te above ideas to te approximation of iger-dimensional surfaces. A crucial point is to construct numerical scemes wic reflect te intrinsic nature of te flow. Section 5 is concerned wit graps. We prove an error bound for a semidiscrete finite element sceme tereby sowing te virtue of working wit geometric quantities. A fully discrete sceme along wit stability issues is discussed afterwards. In Section 6 we introduce te level set equation as a way of andling topological canges. We briefly discuss te framework of viscosity solutions wic allows a satisfactory existence and uniqueness teory. For numerical purposes it is convenient to regularize te level set equation. We collect some properties of te regularized problem and clarify its formal similarity to te grap setting. Te approximation of mean curvature flow by pase field metods is considered in Section 7. Even before numerical discretization tere is te notion of approximation of a sarp interface by a diffuse interface of widt O(ɛ). Te pase field approac depends on te notion of a diffuse interfacial energy composed of quadratic gradient and omogeneous free energy terms involving a pase field function. Te coice of double well energy potential is discussed. We recall some analytical results as well as a convergence analysis for a discretization in space by linear finite elements. We finis tis section by discussing te discretization in time togeter wit te question of stability. In Section 8 we introduce te concept of te anisotropy γ togeter wit its relevant

12 12 K. Deckelnick, G. Dziuk and C. M. Elliott properties and subsequently generalize te ideas of te previous sections to tis setting. Finally, Section 9 is concerned wit fourt order flows: we present discretization tecniques for bot surface diffusion and Willmore flow. For te convenience of te reader we ave included a long list of references, wic are related to te subject of tese notes, but not all of wic are cited in te text. 2. Some geometric analysis Te aim of tis section is to collect some useful definitions and results from differential geometry. We refer to Gilbarg and Trudinger (1998) and Giga (2002) for a more detailed exposition of tis material Hypersurfaces A subset Γ R n+1 is called a C 2 -ypersurface if for eac point x 0 Γ tere exists an open set U R n+1 containing x 0 and a function u C 2 (U) suc tat U Γ={x U u(x) =0}, and u(x) 0 forallx U Γ. (2.1) Te tangent space T x Γ is ten te n-dimensional linear subspace of R n+1 tat is ortogonal to u(x). It is independent of te particular coice of function u wic is used to describe Γ. A C 2 -ypersurface Γ R n+1 is called orientable if tere exists a vectorfield ν C 1 (Γ, R n+1 )(i.e., ν C 1 in an open neigbourood of Γ) suc tat ν(x) T x Γand ν(x) =1for all x Γ. In wat follows, we sall assume tat Γ R n+1 is an oriented C 2 -ypersurface. We define te tangential gradient of a function f, wic is differentiable in an open neigbourood of Γ by Γ f(x) = f(x) f(x) ν(x) ν(x), x Γ. Here denotes te usual gradient in R n+1. Note also tat Γ f(x) is te ortogonal projection of f(x) ontot x Γ. It is straigtforward to sow tat Γ f only depends on te values of f on Γ. We use te notation Γ f(x) =(D 1 f(x),...,d n+1 f(x)) (2.2) for te n + 1 components of te tangential gradient. Obviously Γ f(x) ν(x) =0, x Γ. If f is twice differentiable in an open neigbourood of Γ, ten we define

13 Computation of geometric PDEs and mean curvature flow 13 te Laplace Beltrami operator of f as n+1 Γ f(x) = Γ Γ f(x) = D i D i f(x), x Γ. (2.3) 2.2. Oriented distance function A useful level set representation of a ypersurface can be obtained wit te elp of te distance function. Let Γ be as above and assume in addition tat Γ is compact. Te Jordan Brouwer decomposition teorem ten implies tat tere exists an open bounded set Ω R n+1 suc tat Γ = Ω. We assume tat te unit normal field to Γ points away from Ω and define te oriented (signed) distance function d by dist(x, Γ), x R n+1 \ Ω d(x) = 0, x Γ dist(x, Γ), x Ω. It is well known tat d is globally Lipscitz-continuous and tat tere exists δ>0 suc tat i=1 d C 2 (Γ δ ), were Γ δ = {x R n+1 d(x) <δ}. (2.4) Every point x Γ δ can be uniquely written as x = a(x)+d(x)ν(a(x)), x Γ δ, (2.5) were a(x) Γ. Furtermore, d(x) =ν(a(x)),x Γ δ, wic implies in particular tat d(x) 1 in Γ δ. (2.6) Figure 2.1. Grap (rigt) of te oriented distance function for te curve (left).

14 14 K. Deckelnick, G. Dziuk and C. M. Elliott 2.3. Mean curvature Let us next turn to te notion of mean curvature. By assumption, ν is C 1 in a neigbourood of Γ so tat we may introduce te matrix H jk (x) =D j ν k (x), j,k =1,...,n+1,x Γ. (2.7) It is not difficult to sow tat (H jk (x)) is symmetric. Furtermore, n+1 n+1 H jk (x)ν k (x) = D j ν k (x)ν k (x) = 1 2 D j ν 2 (x) =0, k=1 k=1 since ν =1onΓ. Tus,(H jk (x)) as one eigenvalue wic is equal to zero wit corresponding eigenvector ν(x). Te remaining n eigenvalues κ 1 (x),...,κ n (x) are called te principal curvatures of Γ at te point x. We now define te mean curvature of Γ at x as te trace of te matrix (H jk (x)), tat is, n+1 n H(x) = H jj (x) = κ j (x). (2.8) j=1 Note tat (2.8) differs from te more common definition H = 1 n+1 n j=1 H jj. From (2.7) we derive te following expression for mean curvature, H(x) = Γ ν(x), x Γ, (2.9) were Γ f = n+1 j=1 D jf j denotes te tangential divergence of a vectorfield f. In particular we see tat H>0ifΓ=S n and te unit normal field is cosen to point away from S n, i.e., ν(x) =x. Wile te sign of H depends on te coice of te normal ν, te mean curvature vector Hν is an invariant. A useful formula for tis quantity can be obtained by coosing f(x) =x j,j {1,...,n+1} in (2.3) and observing tat D i x j = δ ij ν j ν i. We ten deduce wit te elp of (2.9) tat n+1 Γ x j = D i (ν j ν i )= ( Γ ν)ν j Γ ν j ν = Hν j, so tat i=1 j=1 Γ x = Hν on Γ. (2.10) Letusnextfixapoint x Γ and calculate H( x) for various representations of te surface Γ near x. Level set representation. Suppose tat Γ is given as in (2.1) near x. Clearly, we ten ave ν(x) =± u(x) u(x)

15 Computation of geometric PDEs and mean curvature flow 15 for x U Γ. If te plus sign applies we obtain H = Γ u u = u u = 1 u n+1 i,j=1 ( δ ij u x i u xj u 2 ) u xi x j. (2.11) In te special case tat u(x) =d(x), were d is te oriented distance function to Γ, we obtain in view of (2.6) Grap representation. Suppose tat H(x) = d(x), x Γ. (2.12) U Γ={(x, v(x)) x Ω}, were Ω R n is open, x =(x 1,...,x n )andv C 2 (Ω). Defining u(x, x n+1 ) = v(x) x n+1 we see tat U Γ is te zero level set of u and te above considerations imply tat ( ) v(x) H(x, v(x)) =, (x, v(x)) U Γ, (2.13) 1+ v(x) 2 were is te gradient in R n and te unit normal is cosen as ν = ( v, 1) 1+ v 2. Parametric representation Suppose tat tere exists an open set V R n and a mapping X C 2 (V,R n+1 ) suc tat U Γ=X(V ), rank DX(θ) =n for all θ V. Te vectors X θ 1 (θ),..., X θ n (θ) ten form a basis of T x Γatx = X(θ). We define te metric on Γ by g ij (θ) = X (θ) X (θ), i,j =1,...,n θ i θ j and let g ij be te components of te inverse matrix of (g ij ). We ten ave te following formulae for te tangential gradient of a function f (defined in a neigbourood of Γ) and te mean curvature vector Hν: were g =det(g ij ). n ij (f X) X Γ f = g, (2.14) θ i,j=1 j θ i Hν = 1 n ( g ij g X ) (2.15) g θ i θ j i,j=1

16 16 K. Deckelnick, G. Dziuk and C. M. Elliott 2.4. Integration by parts Let us assume in tis section tat Γ is in addition compact. Te formula for integration by parts on Γ is (cf. Gilbarg and Trudinger (1998)) D i f da = fhν i da i =1,...,n+1, (2.16) Γ Γ were da denotes te area element on Γ and f is continuously differentiable in a neigbourood of Γ. Applying (2.16) wit = fd i g, summing from i =1,...,n+1 and taking into account tat Γ ν i ν = 0, we obtain Green s formula, Γ f Γ g da = f Γ g da. (2.17) Γ Γ In particular, we deduce from (2.10) Hν φ da = Γ x Γ φ da, (2.18) Γ were φ is continuously differentiable in a neigbourood of Γ wit values in R n+1 and Γ x Γ φ = n+1 i=1 Γx i Γ φ i. Tis relation will be very important for te numerical treatment of mean curvature flow. Te above formulae can be generalized to surfaces wit boundaries by including an appropriate integral over Γ Moving surfaces In tis section we sall be concerned wit surfaces tat evolve in time. A family (Γ(t)) t (0,T ) is called a C 2,1 -family of ypersurfaces if, for eac point (x 0,t 0 ) R n+1 (0,T)witx 0 Γ(t 0 ), tere exists an open set U R n+1, δ>0 and a function u C 2,1 (U (t 0 δ, t 0 + δ)) suc tat U Γ(t) ={x U u(x, t) =0} and u(x, t) 0,x U Γ(t). (2.19) Suppose in addition tat eac Γ(t) is oriented by a unit normal field ν(,t) C 1 (Γ(t), R n+1 ) and tat ν C 0 ( 0<t<T Γ(t) {t}, Rn+1 ). Te normal velocity at a point (x 0,t 0 )(x 0 Γ(t 0 )) is ten defined as V (x 0,t 0 )=φ (t 0 ) ν(x 0,t 0 ), were φ C 1 ((t 0 ɛ, t 0 + ɛ), R n+1 ) satisfies φ(t 0 )=x 0 and φ(t) Γ(t) for t t 0 <ɛ. It can be sown tat V (x 0,t 0 ) is independent of te particular coice of φ. Let us calculate V (x 0,t 0 ) for various representations of Γ(t). Level set representation. Let u be as in (2.19); as above we ten ave ν = ± u u. If te plus sign applies and φ C1 ((t 0 ɛ, t 0 + ɛ), R n+1 ) satisfies φ(t 0 )=x 0 as well as φ(t) Γ(t) for t t 0 <ɛ,weave 0= d dt u(φ(t),t)= u(φ(t),t) φ (t)+u t (φ(t),t), Γ

17 Computation of geometric PDEs and mean curvature flow 17 and ence V (x 0,t 0 )= u t(x 0,t 0 ) u(x 0,t 0 ). (2.20) Grap representation. Suppose tat U Γ(t) ={(x, v(x, t)) x Ω}, were Ω R n is open and v C 2,1 (Ω (t 0 δ, t 0 + δ)). Applying te formula for te level set case to u(x, x n+1,t)=v(x, t) x n+1,weobtain for te unit normal field ν = v t V = 1+ v 2 ( v, 1) 1+ v 2. (2.21) 2.6. Transport teorem for integrals Consider a family (Γ(t)) t (0,T ) of evolving ypersurfaces wic satisfies te assumptions made above and suppose in addition tat eac surface Γ(t) is compact. We are interested in te time derivative of certain volume and area integrals. Lemma 2.1. Let g C 1 (Q), were Q is an open set containing Γ(t) {t}. 0<t<T Suppose in addition tat eac surface Γ(t) is te boundary of an open bounded subset Ω(t) R n+1.ten d g g dx = dt Ω(t) Ω(t) t dx + gv da, (2.22) Γ(t) d g g da = dt Γ(t) Γ(t) t da + g gv H da + V da. (2.23) Γ(t) Γ(t) ν Proof. See te Appendix. 3. Definition and elementary properties of mean curvature flow Te purpose of tis section is to introduce motion by mean curvature and to describe some basic features of tis flow. Consider a C 2,1 -family of ypersurfaces (Γ(t)) t [0,T ] R n+1 togeter wit a coice ν of a unit normal. Definition 1. We say tat (Γ(t)) t [0,T ] moves by mean curvature if V = H on Γ(t). (3.1)

18 18 K. Deckelnick, G. Dziuk and C. M. Elliott Here, V denotes te velocity of Γ(t) in te direction of ν and H is mean curvature. As we sall see later, te above equation gives rise to a parabolic equation, or a parabolic system, for te function(s) describing te surfaces Γ(t), to wic an initial condition Γ(0) = Γ 0 (3.2) as to be added. If Γ(t) as a boundary, ten also suitable boundary conditions need to be specified. In order to give a first idea of tis flow we look at te well-known example of te srinking spere. Let Γ(t) = B r(t) (x 0 ) R n+1, oriented by te unit n r(t) outer normal ν(x) = x x 0 r(t). Ten, V = r (t), H = moves by mean curvature provided tat r (t) = n r(t) on Γ(t), so tat Γ(t). Te solution of tis ODE is given by r(t) = r0 2 2nt, 0 t< r2 0 2n, were Γ 0 = B r0 (x 0 ). Note tat Γ(t) srinks to a point as t r2 0 2n. Te main feature of mean curvature flow is its area-decreasing property, wic is a consequence of te following result. Lemma 3.1. Let Γ(t) be a family of evolving ypersurfaces satisfying V = H on Γ(t) and assume tat eac Γ(t) is compact. Ten V 2 da + d Γ(t) =0, dt were Γ is te area of Γ. Γ(t) Proof. Tis follows immediately from coosing g 1 in (2.23) and te evolution law (3.1). Since te law (3.1) gives rise to a second order parabolic problem we expect existence of a smoot solution locally in time for a smoot initial ypersurface Γ 0. Furtermore, maximum and comparison principles are available wic can be used to sow tat two smoot compact solutions wic are initially disjoint will stay disjoint (see, e.g., Ecker (2002)). Using te srinking spere as a comparison solution, it follows in particular tat if Γ(t), 0 t<t is a smoot solution wit Γ 0 B r0 (x 0 ), ten Γ(t) B r0 2 2nt(x 0)for0 t<min(t, r2 0 2n ). In general, solutions will develop singularities in finite time before tey disappear, but tere are certain initial configurations for wic tey stay smoot until tey srink to a point. Teorem 3.2. Let n 2 and assume tat Γ 0 R n+1 is a smoot, compact and uniformly convex ypersurface. Ten (3.1) and (3.2) ave a smoot solution on a finite time interval [0,T) and te Γ(t) converge to a

19 Computation of geometric PDEs and mean curvature flow 19 point as t T. If one rescales te surfaces in suc a way tat te enclosed volume remains fixed, one as convergence against a spere as t T. Proof. See Huisken (1984). Te case n = 1 is usually referred to as curve sortening flow. Teorem 3.3. Assume tat Γ 0 R 2 is a smoot embedded closed curve. Ten (3.1) and (3.2) ave a smoot embedded solution on a finite time interval [0,T), wic srinks to a round point as t T. Proof. Gage and Hamilton (1986) proved tis result for convex Γ 0 ; subsequently Grayson (1987) sowed tat a smoot embedded closed curve remains smoot and embedded and becomes convex in finite time. If te initial curve is not embedded, cusp-like singularities may develop (see Figures 4.2 and 4.3). Te papers of Angenent (1991), Altsculer and Grayson (1992) and Deckelnick (1997) propose various metods of ow to continue te solution past suc a singularity. Te analogue of Teorem 3.3 for surfaces does not old, as can be seen by coosing a suitable dumbbellsaped initial surface wic develops a pinc-off singularity before it srinks to a point (see Figure 4.5 and Grayson (1989)). Tis pinc-off leads to a cange of te topological type of Γ(t), so tat te parametric approac in wic te topological type is fixed will develop a singularity tat is difficult to andle. Tus te question arises weter it is possible to introduce a notion of solution tat is capable of following te flow troug a singularity. Several suc notions ave been proposed and analysed starting wit te pioneering work of Brakke (1978) on varifold solutions, wic uses tools from geometric measure teory. In tis context we also mention te surface evolver program of Brakke (1992). Level set and pase field metods constitute two completely different approaces wic take an Eulerian point of view. We sall discuss tese in more detail in Sections 6 and Parametric mean curvature flow As is mentioned above, in te parametric approac one cooses a suitable reference manifold M R n+1 (of te topological type of te evolving ypersurfaces Γ(t)) and ten looks for maps X(,t):M R n+1 (0 t<t)suc tat Γ(t) =X(,t)(M). To fix ideas, let us assume tat M is a compact ypersurface witout boundary. If we can find X in suc a way tat X (p, t) = H(X(p, t))ν(x(p, t)) (p, t) M (0,T), (4.1) t ten V = H on Γ(t) follows by taking te dot product wit te normal ν. In order to understand (4.1) let F :Ω R n+1 be a local parametrization

20 20 K. Deckelnick, G. Dziuk and C. M. Elliott of M definedonanopensetω R n and set ˆX(θ, t) =X(F (θ),t), (θ, t) Ω (0,T). Recalling (2.15), te equation (4.1) ten turns into ˆX 1 n ( (θ, t) ĝ ĝ ij ĝ ˆX ) =0 (4.2) t θ i θ j i,j=1 were ĝ ij (θ, t) = ˆX θ i ˆX θ j and ĝ ij, ĝ are as above. Tus (4.2), and ence (4.1), is a nonlinear parabolic system, wic is not defined at points (θ, t) were ĝ(θ, t) = 0. In order to close tis system, an initial condition X(p, 0) = X 0 (p),p M needs to be prescribed, were X 0 : M R n+1 is a parametrization of te initial surface Γ Curve sortening flow Mean curvature evolution in te one-dimensional case is usually referred to as curve sortening flow. In te case of closed curves, a convenient coice of a reference manifold is M = S 1, wic can be parametrized globally by F (θ) =(cosθ, sin θ),θ [0, 2π]. In te following, for simplicity, let us identify ˆX(θ, t) andx((cos θ, sin θ),t). Tus, (4.2) becomes X t 1 ( ) Xθ =0 X θ X θ θ ini (0,T), (4.3) X(, 0) = X 0 in I, (4.4) were I =[0, 2π]. In addition, X as to satisfy te periodicity condition X(θ, t) =X(θ +2π, t) 0 t<t, θ R. (4.5) Suppose tat X : R [0,T] R 2 is a smoot solution of (4.3) (4.5), in particular X θ > 0inI [0,T]. If we multiply (4.3) by X θ, take te dot product wit a test function ϕ Hper(I; 1 R 2 )={ϕ H 1 (I; R 2 ) ϕ(0) = ϕ(2π)} and integrate over I, we obtain X θ ϕ θ X t ϕ X θ + =0 forallϕ H 1 I I X θ per(i; R 2 ). (4.6) We use (4.6) in order to discretize in space. For simplicity let θ j = j (j =0,...,N) be a uniform grid wit grid size =2π/N and let { } S = ϕ C 0 (I; R 2 ) ϕ [θj 1,θ j ] P1 2,j =1,...,N; ϕ (0) = ϕ (2π) be te space of piecewise linear continuous functions wit values in R 2.Te spatial discretization of (4.3) is ten given by X θ ϕ θ X t ϕ X θ + =0 forallϕ S. (4.7) X θ I I

21 Computation of geometric PDEs and mean curvature flow 21 Denoting te common (scalar) nodal basis by {φ 1,...,φ N }, we can expand X (θ, t) = N j=1 X j(t)φ j (θ) wit vectors X j (t) R 2. Tis one-dimensional finite element formulation can be rewritten as a difference sceme. To see tis, insert ϕ = φ j e k, (k =1, 2; j =1,...,N) into (4.7) and calculate X t ϕ X θ dθ = 1 6 X j X j 1 Ẋj 1 e k I as well as ( X j X j 1 + X j+1 X j )Ẋj e k X j+1 X j Ẋj+1 e k I X θ ϕ θ X θ dθ = ( X j+1 X j q j+1 + X ) j X j 1 e k. q j Here, q j = X j X j 1 and te dot stands for te time derivative. Tus, (4.7) can be written as 1 6 q jẋj (q j + q j+1 )Ẋj q j+1ẋj+1 = X j+1 X j X j X j 1 (4.8) q j+1 q j (j =1,...,N). If we use mass lumping in (4.8) we get te difference sceme 1 2 (q j + q j+1 )Ẋj = X j+1 X j q j+1 As initial values for X j we coose X j X j 1 q j. (4.9) X j (0) = X 0 (θ j ), j =0,...,N, (4.10) so tat X (, 0) is te linear interpolant of X 0. Furtermore we require te periodicity condition X j = X j+n, j = 1, 0, 1. (4.11) Te following proposition sows tat te lumped sceme reflects te curve sortening property of te exact solution. Proposition 4.1. Consider solutions X of (4.3) and X = N j=1 X j(t)φ j (θ) of (4.9) respectively. Ten we ave for t [0,T] X θ (,t) t = X t (,t) 2 X θ (,t) in I q j = 1 4 (q j 1 + q j ) Ẋj (q j + q j+1 ) Ẋj 2, j =1,...,N as long as q j > 0,j =1,...,N. Tus, te faces of te polygon wit vertices X 1,...,X N decrease in lengt during time evolution. Proof. For te proof of te first assertion we differentiate X θ 1 twice wit respect to θ and get ( ) X θ X θ Xθ =0, X θ θ ( Xθ ) 2 = X ( ) θ X θ X θ Xθ X θ θ X θ θθ in I,

22 22 K. Deckelnick, G. Dziuk and C. M. Elliott wic combined wit (4.3) gives X θ t = X θ X θ X θt = X ( ) θ X θ 1 Xθ X θ X θ θθ = 1 ( ) Xθ 2 = X t 2 X θ. X θ X θ θ For te discrete solution we observe tat by (4.9) wit te unit vectors T j = X j X j 1 q j we ave q j = T j (Ẋj ( Ẋj 1) ) 2 2 = T j (T j+1 T j ) (T j T j 1 ) q j + q j+1 q j 1 + q j 2 2 = (1 T j T j+1 ) (1 T j 1 T j ) q j + q j+1 q j 1 + q j 1 = T j T j ( T j 1 T j 2 q j + q j+1 q j 1 + q j = 1 4 (q j 1 + q j ) Ẋj (q j + q j+1 ) Ẋj 2. For tis we ave used te discrete equation (4.9) twice. Under te assumption tat a smoot and regular solution of te curve sortening flow (4.3) (4.5) exists, one obtains te following convergence result togeter wit error estimates for te position vector X and te velocity vector X t, wic by (4.1) is equal to te curvature vector. Te proof follows from Dziuk (1994) and is a special case of Teorem 8.4. Teorem 4.2. Let X : I [0,T] R 2 be a periodic smoot solution of te curve sortening flow (4.3) (4.5) wit X θ c 0 > 0inI [0,T]. Ten tere exists an 0 > 0 depending on X and T suc tat for every 0 < 0 tere exists a unique solution X (θ, t) = N j=1 X j(t)φ j (θ) of te difference sceme (4.9), (4.10) and ( T ) 1/2 max X X L [0,T ] 2 (I) + X θ X θ 2 L 2 (I) dt c, (4.12) 0 ( T ) 1/2 max X t X t L [0,T ] 2 (I) + X tθ X tθ 2 L 2 (I) dt c, (4.13) 0 were c depends on X and T. Tis algoritm can be generalized witout canges to curves evolving in iger codimension, i.e., X : I [0,T] R m and m>2. Te curve solving (4.3) as a velocity only in te normal direction. It is also possible to use

23 Computation of geometric PDEs and mean curvature flow 23 te parametric equation X t = X θθ X θ 2 instead, wic defines te same curve evolving in te normal direction wit a normal velocity being given by te curvature. However, te parametrization is different, wit te points on te curve now aving a tangential velocity. A finite element error analysis for te motion of a closed curve is given in Deckelnick and Dziuk (1994), wile error bounds for te evolution of a curve attaced to a fixed boundary wit a normal contact condition are proved in Deckelnick and Elliott (1998). In order to obtain a practical metod we still ave to discretize in time. Coose a time-step τ>0 and let t m = mτ, m =0,...,M, M [ T τ ]. We let X m S denote te approximation to X(,t m ). On te basis of (4.7) we suggest te following sceme: 1 (X m+1 ϕ θ τ I I Xθ m =0 forallϕ S. (4.14) Calculations similar to tose above yield a time discrete analogue of (4.9), wic we formulate as te following algoritm. X m ) ϕ X m θ + X m+1 θ Algoritm 1. (Curve sortening flow) (1) Let Xj 0 = X 0(θ j )(j =0,...,N). (2) Compute Xj m+1 (j = 0,...,N) from te tridiagonal systems ( 1 X m+1 2τ (qm j +qj+1)(x m j m+1 Xj m j+1 ) Xm+1 j qj+1 m Xm+1 j Xj 1 m+1 ) qj m =0. (3) If min j=1,...,n+1 qj m+1 > 0 ten replace m by m +1andgoto 2. Tus, in eac time-step a positive definite and symmetric linear system as to be solved for eac component of X m+1. Eac of tese linear systems is of tridiagonal form wit two additional entries reflecting te periodicity condition. Te system decouples wit respect to te dimension of te space in wic te curve moves. For practical purposes a redistribution of nodes according to arc lengt on te curve is sometimes convenient. Let us go back to te more precise notation ˆX(θ, t) =X((cos θ, sin θ),t). For later purposes it is convenient to look at (4.14) from a sligtly different angle. We introduce te polygon Γ m = ˆX m (I) along wit te space S m = {φ :Γ m R2 φ is affine on eac face of Γ m }. (4.15) Tus, if φ S m, ten φ is te restriction of an affine function on R 2 on eac face of te polygon and terefore ϕ (θ) =φ ( ˆX m (θ)), θ I,

24 24 K. Deckelnick, G. Dziuk and C. M. Elliott Figure 4.1. Curve sortening flow applied to a star-saped curve. Time-steps 0, 100, 200, 300, 500, 700, 5000, 7000 (time-step size = ), 480 nodes. Figure 4.2. Curve sortening flow applied to a curve wit a self-intersection. A singularity (cusp) appears. Te effect is tat te algoritm jumps across te singularity. See Figure 4.3 for a magnified image. Time-steps 0, 1000, 2000, 2500, 3000, 5000, 6000, 7000 (time-step size = ), 480 nodes. Figure 4.3. Close-up of Figure 4.2. Time-steps 3498 and 3499 and Te parametric teory breaks down.

25 Computation of geometric PDEs and mean curvature flow 25 belongs to Ŝ. Recalling (2.14) we ave Γ m φ = 1 ˆX θ m ϕ θ ˆX θ m ˆX θ m, p = ˆX m (θ), were (u v) ij = u i v j (u, v R 2 )and Γ m φ is given piecewise on eac face of Γ m. If we define Xm+1 S m m+1 by Xm+1 (p) = ˆX (θ), p= ˆX m(θ). Observing tat Γ m X m+1 Γ m φ ˆX θ m m+1 ˆX θ ϕ θ = ˆX θ m for all ϕ Ŝ we can rewrite (4.14) as 1 (X m+1 τ id) φ da+ Γ m Γ m Γ m X m+1 Γ m φ da =0 forallφ S m. (4.16) Note tat te dot between te matrices Γ m X m+1 and Γ m φ is te standard scalar product in R 4. Te key point about te formulation (4.16) is tat Γ m+1 is now parametrized wit te elp of te polygon Γ m from te previous time-step, so tat te reference manifold M is no longer needed. We can interpret te second integral on te left-and side of (4.16) as an approximation to Γ(tm+1 )x Γ(tm+1 )φ da, Γ(t m+1 ) wic equals Γ(t m+1 ) Hν φ da by (2.17) and (2.10). Here, H is just te usual curvature of te curve Γ(t m+1 ), but of course it is now natural to also use (4.16) for approximating surfaces evolving by mean curvature. We will discuss tis issue in te next section Mean curvature flow of surfaces In tis section we sall use a iger-dimensional version of (4.16) in order to approximate parametric surfaces Γ(t) = X(M,t), wic satisfy (4.1). To begin, we need an analogue of te polygons used in te previous section. Figure 4.4. Polyedral surfaces: successively refined grids approximating a alf spere. Macro triangulation (left) and triangulation levels 1, 5 and 7.

26 26 K. Deckelnick, G. Dziuk and C. M. Elliott Figure 4.5. First row: Parametric mean curvature flow of a dumbbell-saped surface. Development of a singularity. Second row: Axially symmetric level set computation of te same flow going beyond te topological cange of te surface. Definition 2. We call a set Γ R n+1 a polyedral surface if Γ= T, T T were te triangulation T consists of closed, nondegenerate, n-dimensional simplices. Te intersection of two adjacent simplices is an (n k)-dimensional subsimplex of tese simplices (k {1,...,n}). Our aim is to construct polyedral surfaces Γ 0,...,ΓM (witout boundary) in suc a way tat Γ m is an approximation to Γ(t m). Tese surfaces are obtained wit te elp of te following algoritm. We start te computations wit an initial polyedral Γ 0 wic approximates te initial surface Γ 0. In practice tere are several ways to construct te initial discrete surface. One way is to map triangulations of carts onto te continuous surface and to glue tem togeter. A muc better way is to construct a macro triangulation, tat is, a coarse approximation Γ 0 of Γ 0 suc tat

27 Computation of geometric PDEs and mean curvature flow 27 Figure 4.6. A tin two-dimensional torus srinks under parametric mean curvature flow to a circle. Figure 4.7. A tick two-dimensional torus (cut open) srinks under parametric mean curvature flow to a spere developing a singularity. Γ 0 Γδ (see (2.4), (2.5)) and ten to refine tis triangulation in Rn+1 and project te new nodes x ortogonally onto te smoot surface according to x = x d(x)ν(x) to obtain te new nodes x of te next-finer triangulation for Γ0 (see Figure 4.4). Algoritm 2. (Mean curvature flow of surfaces) Let Γ0 be a polyedral approximation of Γ0. For m = 0, 1,..., M 1 define m Sm = {φ C 0 (Γm ) φ T is affine for eac T Γ }, and find Xm+1 Sm wit 1 (Xm+1 id)φ da + Γm Xm+1 Γm φ da = 0 m τ Γm Γ for all φ Sm (4.17) m+1 m = X (Γ ), and if it is a polyedral Generate te new surface Γm+1 surface ten goto to te next m.

28 28 K. Deckelnick, G. Dziuk and C. M. Elliott Tis algoritm is based on a finite element metod for partial differential equations on surfaces, developed in Dziuk (1988). Let us ave a look at te implementation of te above algoritm. Fix m {0,...,M 1} and denote by a 1,...,a N R n+1 te nodes of te polyedral surface Γ m. Te functions φ i :Γ m R, i=1,...,n are uniquely defined by te requirements φ i S m, φ i(a j )=δ ij, i,j =1,...,N. It is not difficult to verify tat φ 1,...,φ N actually form a basis of S m.now, stiffness and mass matrix are defined by S ij = Γ m φ i Γ m φ j da, i, j =1,...,N M ij = Γ m Γ m φ i φ j da, i, j =1,...,N. Expanding (X m+1 ) k (p) = N j=1 α(k) j φ j (p) (were (X m+1 ) k is te kt component of X m+1 ), we find tat (4.17) is equivalent to te linear systems Mα (k) + τsα (k) = b (k), k =1,...,n+1. (4.18) Here, α (k) =(α (k) 1,...,α(k) N )andb(k) R N is given by b (k) j = x k φ j da, j =1,...,N. Γ m Since te matrix M + τs is symmetric and positive definite, te systems (4.18) can be solved wit a conjugate gradient metod. Te only difference to a Cartesian FEM is tat te nodes ave one more coordinate. 5. Mean curvature flow of graps We turn our attention to te mean curvature evolution of surfaces Γ(t), wic can be written as graps over some base domain Ω R n, tat is, Γ(t) ={(x, u(x, t)) x Ω}. In order to find te differential equation to be satisfied by te eigt function u, we recall (2.13) and (2.21) to see tat te mean curvature H and te velocity V in te direction of ν = ( u, 1) are given by 1+ u 2 ( ) u u H =, V = t. (5.1) 1+ u 2 1+ u 2

29 Computation of geometric PDEs and mean curvature flow 29 Tus, te evolution law V = H on Γ(t) translates into te nonlinear parabolic partial differential equation u t ( ) 1+ u 2 u =0 inω (0,T), (5.2) 1+ u 2 to wic we add te following boundary and initial conditions u = g on Ω (0,T), (5.3) u(, 0) = u 0 in Ω, (5.4) were g : Ω R and u 0 : Ω R are given functions. Te boundary condition (5.3) implies tat te boundaries of te surfaces Γ(t) arekept fixed during te evolution. It would also be possible to replace (5.3) by u =0 n on Ω (0,T), (5.5) in wic case te surfaces Γ(t) would meet te boundary of te cylinder Ω R at a rigt angle Analytical results Te main difficulties for te matematical analysis are due to te fact tat te operator A(u) = ( ) 1+ u 2 u 1+ u 2 is not uniformly parabolic and not in divergence form. Only in one space dimension te equation is in divergence form, since A(u) = (arctan u x ) x. Teorem 5.1. Let Ω be a bounded domain in R n wit Ω C 2+α and u 0 C 2,α ( Ω). (a) Suppose tat g C 2,α ( Ω) and tat te compatibility conditions u 0 = g and ( ) 1+ u 0 2 u 0 =0 on Ω 1+ u0 2 are satisfied. If Ω as nonnegative mean curvature, te initial-boundary value problem (5.2), (5.3), (5.4) as a unique smoot solution wic converges to te solution of te minimal surface equation wit boundary data g as t. (b) Suppose tat te compatibility condition u 0 n =0on Ω olds. Ten te initial-boundary value problem (5.2), (5.5), (5.4) as a unique smoot solution wic converges to a constant function as t. Proof. See Lieberman (1986) and also Huisken (1989) for (a); (b) is proved in Huisken (1989).

30 30 K. Deckelnick, G. Dziuk and C. M. Elliott Te assumption tat te boundary of te domain as nonnegative mean curvature is a necessary condition. If it is dropped, te gradient of te solution will become infinite on te boundary: see Oliker and Uraltseva (1993). Te main tool in te proof of te previous teorem is te derivation of an evolution equation for te surface element. Our numerical algoritms will be based on a variational formulation of (5.2), (5.3). To derive it, divide (5.2) by Q = 1+ u 2, (5.6) multiply by a test function φ H0 1 (Ω) and integrate. Integration by parts implies u t φ Ω Q + u φ =0, φ H0 1 (Ω), 0 <t<t. (5.7) Ω Q It is straigtforward to derive from (5.7) te decrease in area. Lemma 5.2. Suppose tat u is a smoot solution of (5.2). Ten u 2 t Ω Q + d Q =0. (5.8) dt Ω Proof. Since u(,t)=g on Ω (0,T)weaveu t (,t)=0on Ω for 0 <t<t. Te relation (5.8) now follows by inserting φ = u t (,t) in (5.7) and observing tat Q t = u ut Q. Recalling tat V = ut Q we may rewrite te relation (5.8) in te more geometric form of Lemma Spatial discretization Let T be an admissible nondegenerate triangulation of te domain Ω wit mes size bounded by, simplices S and Ω = S T S te corresponding discrete domain. We assume tat vertices on Ω are contained in Ω. Te space of finite elements of order s N is cosen to be X = {v C 0 (Ω ) v is a polynomial of order s on eac S T }. (5.9) Te subspace containing functions wit zero boundary values will be denoted by X 0. We assume tat for s N,p [1, ] tere exists an interpolation operator I : H s+1,p (Ω) X wic satisfies I v X 0 for v H s+1,p (Ω) H 1 0 (Ω), as well as v I v L p (Ω Ω ) + (v I v) L p (Ω Ω ) c s+1 v H s+1,p (Ω) (5.10) for all v H s+1,p (Ω). For dimensions n<p(s + 1), we can, for instance, coose te usual Lagrange interpolation operator; in iger dimensions a

31 Computation of geometric PDEs and mean curvature flow 31 possible coice is te Clément operator. For wat follows we coose piecewise linear finite elements: s =1. We now use (5.7) in order to define a semidiscrete approximation to te solution of (5.2) (5.4) as follows: find u (,t) X wit u (,t) I g X 0 and u (, 0) = u 0 = I u 0 suc tat Ω u t φ Q + Ω u φ Q =0, for all φ X 0 (5.11) and all t (0,T). Here, we ave abbreviated Q = 1+ u 2. Te following lemma establises te global existence of te discrete solution. Lemma 5.3. Te semidiscrete problem as a unique solution u wic exists globally in time. Proof. We denote by a i,i=1,...,n te nodes of te triangulation T and by χ i te corresponding nodal basis functions. We assume tat a 1,...,a N1 are te interior nodes, wile a N1 +1,...,a N lie on Ω. We expand u (.t) = N1 i=1 α i(t)χ i + N i=n 1 +1 g(a i)χ i and te relation (5.11) ten amounts to a nonlinear system of ODEs for α =(α 1,...,α N1 ). Existence of a unique local solution follows from standard ODE teory, wile te analogue of (5.8) implies a uniform bound on u and terefore on α since X is finitedimensional. Tis allows us to continue te solution for all times. In order to prove error estimates for te semidiscrete problem we need to make regularity assumptions on te solution of te continuous problem. Let us suppose tat u satisfies T 0 u t 2 H 1, (Ω) dt + T 0 u t 2 H 2 (Ω) dt N (5.12) for some N>0 (see Deckelnick and Dziuk (1999) for sufficient conditions wic imply (5.12)). In te following we sall assume tat we ave a solution of tis kind until te time T. We sall formulate our error estimates in terms of geometric quantities, more specifically in terms of te normals ν = ( u, 1) Q,ν = ( u, 1) Q and te normal velocities V = ut Q,V = u t Q reflecting te form of te aprioriestimate (5.8). Teorem 5.4. Let u be a solution of te continuous problem (5.2) (5.4), wic satisfies (5.12). Ten T (V V )) 2 Q +sup ν ν 2 Q c 2. 0 Ω Ω (0,T ) Ω Ω Te constant c depends on N.

32 32 K. Deckelnick, G. Dziuk and C. M. Elliott Proof. Let us give te proof of tis teorem for polygonal domains, Ω = Ω. Te proof sows ow important it is to work wit te geometric quantities. Te difference of te discrete weak form (5.11) and te corresponding continuous weak form of equation (5.2) reads Ω ( ut Q u t Q ) φ + Ω ( u Q u Q ) φ = 0 (5.13) for all discrete test functions ϕ X 0. As a test function we coose φ = I u t u t =(u t u t ) (u t I u t ). We observe tat ( ut Q u ) t (u t u t )=(V V )(VQ V Q ) (5.14) Q =(V V ) 2 Q +(V V )V (Q Q ) (V V ) 2 Q V V V Q 1 Q 1 Q Q 1 2 (V V ) 2 Q 1 2 u t 2 ν ν 2 Q. Here we ave used te fact tat 1 Q 1 Q ν ν. (5.15) For te gradient term in (5.13) we exploit te fact tat te last component of te vector νq ν Q is zero, and get ( u Q u ) ( u t u t )=(ν ν ) ( u t u t, 0) (5.16) Q =(ν ν ) (νq ν Q ) t. Wit te elementary relation (ν ν ) ν = (ν ν ) ν = 1 2 ν ν 2, te rigt-and side in (5.16) can be estimated as follows: (ν ν ) (νq ν Q ) t =(ν ν ) (ν t Q ν t Q + νq t ν Q t ) = 1 2 ν ν 2 (Q t + Q t )+(ν ν ) (ν ν ) t Q +(ν ν ) ν t (Q Q ) = 1 2 ( ν ν 2 Q ) t ν ν 2 Q t +(ν ν ) ν t (Q Q ) 1 2 ( ν ν 2 Q ) t 1 2 Q t ν ν 2 ν t Q ν ν 2 Q,

33 Computation of geometric PDEs and mean curvature flow 33 were again we ave used (5.15). Wit tis estimate, (5.14) and (5.16) te error relation (5.13) implies te bound 1 V ) 2 Ω(V 2 Q + 1 d ν ν 2 Q 2 dt Ω 2( 1 ut 2 L (Ω) +3 u t 2 ) L (Ω) ν ν 2 Q Ω + V V u t I u t + ν ν (u t I u t ). Ω We estimate te interpolation terms wit te elp of (5.10), tat is, V V u t I u t + ν ν (u t I u t ) Ω Ω ( ) 1 ( ) 1 ) c u t H 2 (Ω) ( 2 (V V ) ν ν 2 2 Ω Ω δ (V V ) 2 Q + δ ν ν 2 Q + c Ω Ω δ u t 2 H 2 (Ω) 2 for every δ>0, since Q 1. After a suitable coice of δ we arrive at 1 V ) 2 Ω(V 2 Q + d ν ν 2 Q dt Ω c ( 1+ u t 2 ) H 1, (Ω) ν ν 2 Q + c u t 2 H 2 (Ω) 2. Ω A Gronwall argument and te coice u (, 0) = I u 0 ten finally proves te teorem. Remark 1. It is possible to sow tat in te two-dimensional case te above error bounds imply tat sup Ω [0,T ] Q C uniformly in. As a consequence te error estimate can be written down wit te elp of te usual norms, namely T 0 u t u,t 2 L 2 (Ω Ω ) dt +sup (u u ) 2 L 2 (Ω Ω ) c2. (0,T ) Ω 5.3. Time discretization Let us coose a time-step τ>0 and let t m = mτ, m =0,...,M,M [ T τ ] as well as v m = v(,mτ)form =0,...,M. Based on (5.11) we suggest te following algoritm. Algoritm 3. (Mean curvature flow of graps) Let u 0 = I u 0. For m =0,...,M 1, compute u m+1 X suc tat u m+1 I g X 0 and,

34 34 K. Deckelnick, G. Dziuk and C. M. Elliott for every ϕ X 0, 1 τ m+1 ϕ u Ω Q m wit Q m = 1+ u m 2. + Ω u m+1 Q m ϕ = 1 τ Ω u m ϕ Q m. (5.17) Te above sceme is semi-implicit in time and as te property tat in eac time-step a linear Laplace-type equation wit stiffness matrix weigted by Q m as to be solved. In order to analyse its stability we go back to te basic energy norms introduced in (5.8). Teorem 5.5. Te solution u m, 0 m M of (5.17) satisfies, for every m {1,...,M}, m 1 τ V k 2 Q k + Q m Q 0 (5.18) Ω Ω Ω k=0 were V k = (uk+1 u k ) is te discrete normal velocity. τq k Proof. We coose ϕ = u k+1 u k as a test function in (5.17) for m = k and get 1 τ Ω (u k+1 u k )2 Q k + Ω u k+1 (u k+1 Q k u k ) =0. (5.19) Let us use te notation ν k = ( uk, 1). Te integrand in te second term Q k can be rewritten as u k+1 (u k+1 u k ) Q k = (Qk+1 ) 2 1 Q k uk+1 Q k+1 = (Qk+1 ) 2 Q k νk+1 ν k 2 Q k+1 Q k+1 = 1 2 νk+1 ν k 2 Q k+1 + Q k+1 uk Q k Q k + (Qk+1 Q k )2. Q k Q k+1 We insert tis result into (5.19), sum over k =0,...,m 1 and obtain te equation m 1 m 1 τ V k 2 Q k + k+1 (Q Q k )2 k=0 Ω k=0 Ω Q k + 1 m 1 ν k+1 2 ν k 2 Q k+1 k=0 Ω + Q m = Q 0 Ω Ω wic implies te stability estimate (5.18).

35 Computation of geometric PDEs and mean curvature flow 35 Let us empasize tat our sceme is unconditionally stable even toug te nonlinear expressions are treated explicitly. Oter scemes, suc as fully explicit and fully implicit variants are discussed in Dziuk (1999a). It is natural to follow te ideas of te semidiscrete case in order to analyse te above algoritm. For te analysis of te fully discrete sceme we need te following regularity assumptions: ( u(,t) H 2, (Ω) + u t (,t) H (Ω)) 1, sup t (0,T ) + T Tis leads to te following result. 0 ( ut 2 H 2 (Ω) + u tt 2) ds N. (5.20) Teorem 5.6. Assume tat tere exists a solution of (5.2) (5.4) on [0,T], wic satisfies (5.20) and let u m,(m =1,...,M =[T τ ]) be te solution of Algoritm 3. Ten tere exists a τ 0 > 0 suc tat, for all 0 <τ τ 0, M 1 τ (V m V m )2 Q m c(τ ), (5.21) m=0 Ω Ω sup ν m ν m 2 Q m c(τ ). (5.22) m=0,...,m Ω Ω Proof. Tis is a special case of te results obtained in Deckelnick and Dziuk (2002a). For computational tests we refer to te anisotropic case; see Table 8.2. Here we give some test results for te usual norms. Error estimates in tese norms for te two-dimensional case are contained in Deckelnick and Dziuk (2000). For te tests we ave solved te partial differential equation Table 5.1. Absolute errors in L ((0,T); L 2 (Ω)), L 2 ((0,T); H 1 (Ω)) and experimental orders of convergence (EOC) for te test problem. E 1 EOC E 2 EOC

36 36 K. Deckelnick, G. Dziuk and C. M. Elliott (5.2) wit a given additional rigt-and side. We ave cosen u(x, t) = sin ( x 2 t) sin (1 t) and calculated a rigt-and side from tis function. Te computational domain was Ω = {x R 2 x < 1} and we used te boundary condition u =0on Ω. Te time interval was [0, 4] and as timestep size we ave cosen τ = For two successive grids wit grid sizes 1 and 2 we computed te absolute errors E( j ), (j =1, 2) between discrete solution and exact solution for certain norms. Te experimental order of convergence was ten defined by EOC = log (E( 1 )/E( 2 ))/ log ( 1 / 2 ). In Table 5.1 te errors in te norms E 1 = sup 0 m M u m u m wit Mτ = T and E 2 = sup 0 m M (u m u m ) are sown. Te results confirm te teoretical estimates. Note tat te L ((0,T),L 2 (Ω))-error beaves linearly in te grid size because we ave cosen te time-step proportional to te spatial grid size. 6. Mean curvature flow of level sets If we want to compute topological canges of free boundaries ten it is necessary to leave te parametric world, because tis fixes te topological type of te interface. One metod to do tis is to define te interface as te level set of a scalar function: Γ(t) = { x R n+1 u(x, t) =0 }. Let us assume for te moment tat u C 2,1 (R n+1 (0,T)) wit u 0 in a neigbourood of t (0,T ) Γ(t) {t}. Recalling (2.11) and (2.20), te relation V = H on Γ(t) would old if n+1 ( u t δ ij u ) x i u xj u 2 u xi x j =0 (6.1) i,j=1 in a neigbourood of t (0,T ) Γ(t) {t}. Tis partial differential equation is igly nonlinear, degenerate parabolic and not defined were te gradient of u vanises. Terefore, standard metods for parabolic equations fail, but it is possible to develop an existence and uniqueness teory for (6.1) witin te framework of viscosity solutions. Te corresponding notion involves a pointwise relation and te analysis relies mainly on te maximum principle. It is terefore not straigtforward to use finite element metods, wic are typically L 2 -metods and normally do not allow a maximum principle. Tis difficulty will be reflected in te numerical analysis. An example of te evolution of level sets under mean curvature flow is sown in Figure 6.1 (Deckelnick and Dziuk 2001). Crandall, Isii and Lions (1992) give a concise introduction to te teory of viscosity solutions, wile Giga (2002) describes in detail te application of level set tecniques to a large class of geometric evolution equations.

37 Computation of geometric PDEs and mean curvature flow 37 Figure 6.1. Evolution of level lines under mean curvature flow. Detailed descriptions of computational tecniques for level set metods along wit a ost of applications can be found in te monograps by Setian (1999) and Oser and Fedkiw (2003) Analytical results Starting from (6.1), we are interested in te following problem: n+1 ( u t δ ij u ) x i u xj u 2 u xi x j =0 inr n+1 (0, ) (6.2) i,j=1 i,j=1 u(, 0) = u 0 in R n+1. (6.3) An existence and uniqueness teory for (6.2), (6.3) can be carried out witin te framework of viscosity solutions. Definition 3. A function u C 0 (R n+1 [0, )) is called a viscosity subsolution of (6.2) provided tat for eac φ C (R n+2 ), if u φ asalocal maximum at (x 0,t 0 ) R n+1 (0, ), ten n+1 ( φ t δ ij φ ) x i φ xj φ 2 φ xi x j 0 at (x 0,t 0 ), if φ(x 0,t 0 ) 0, (6.4) n+1 φ t (δ ij p i p j )φ xi x j 0 at (x 0,t 0 ) for some p 1, i,j=1 if φ(x 0,t 0 )=0.

38 38 K. Deckelnick, G. Dziuk and C. M. Elliott Figure 6.2. Evolution of a lemniscate under level set mean curvature flow: te zero level. A viscosity supersolution is defined analogously: maximum is replaced by minimum and by. A viscosity solution of (6.2) is a function u C 0 (R n+1 [0, )) tat is bot a subsolution and a supersolution. We sall assume tat te initial function u 0 is smoot and satisfies u 0 (x) =1 for x S (6.5) for some S>0. Te following existence and uniqueness teorem is a special case of results proved independently by Evans and Spruck (1991) and Cen, Giga and Goto (1991). Teorem 6.1. Assume u 0 : R n+1 R satisfies (6.5). Ten tere exists a unique viscosity solution of (6.2), (6.3), suc tat u(x, t) =1 for x + t R for some R>0 depending only on S. Te level set approac can now be described as follows: given a compact ypersurface Γ 0, coose a continuous function u 0 : R n+1 R suc tat Γ 0 = {x R n+1 u 0 (x) =0}. If u : R n+1 [0, ) R is te unique viscosity solution of (6.2), (6.3), we ten call Γ(t) ={x R n+1 u(x, t) =0}, t 0

39 Computation of geometric PDEs and mean curvature flow 39 Figure 6.3. Evolution of te oriented distance function of a lemniscate: level lines. a generalized solution of te mean curvature flow problem. We remark tat Evans and Spruck (1991) and Cen, Giga and Goto (1991) also establised tat te sets Γ(t) ={x R n+1 u(x, t) =0},t > 0 are independent of te particular coice of u 0 wic as Γ 0 as its zero level set, so tat te generalized evolution (Γ(t)) t 0 is well defined for a given Γ 0.AsΓ(t) exists for all times, it provides a notion of solution beyond singularities in te flow. For tis reason, te level set approac as also become very important in te numerical approximation of mean curvature flow and related problems. Note owever tat it is possible tat te set Γ(t) may develop an interior for t>0, even if Γ 0 ad none, a penomenon wic is referred to as fattening. Te level set solution as been investigated furter in several papers: in particular we mention Evans and Spruck (1992a, 1992b, 1995) and Soner (1993) Regularization Evans and Spruck (1991) proved tat te (smoot) solutions u ɛ of n+1 ( u ɛ t δ ij uɛ x i u ɛ ) x j ɛ 2 + u ɛ 2 u ɛ x i x j =0 inr n+1 (0, ), (6.6) i,j=1 u ɛ (, 0) = u 0 in R n+1 (6.7)

40 40 K. Deckelnick, G. Dziuk and C. M. Elliott Figure 6.4. Evolution of te oriented distance function of a lemniscate: grap. converge locally uniformly as ɛ 0 to te unique viscosity solution of (6.2), (6.3). For numerical purposes it is important to know te asymptotic error between te viscosity solution and te solution of te regularized problem quantitatively as ɛ 0. In Deckelnick (2000) tere is a proof of te following teorem togeter wit several aprioriestimates and teir dependence on te regularization parameter ɛ. Teorem 6.2. For every α (0, 1 2 ), 0 < T < tere is a constant C = C(u 0,T,α) suc tat sup u u ɛ L (R n+1 ) Cɛ α for all ɛ>0. 0 t T If one wants to calculate approximations to te viscosity solution u of (6.2), (6.3) ten, according to Teorem 6.2, it is sufficient to solve te regularized problem (6.6), (6.7), wic we ave to study for computational purposes, on a bounded domain. For simplicity we coose Ω = B S(0) wit S >R= R(S), were R is te radius from Teorem 6.1, and consider n+1 u ɛt i,j=1 ( δ ij u ) ɛx i u ɛxj ɛ 2 + u ɛ 2 u ɛxi x j =0 inω (0, ), (6.8) u ɛ =1 on Ω (0, ), (6.9) u ɛ (, 0) = u 0 in Ω. (6.10)

41 Computation of geometric PDEs and mean curvature flow 41 An application of te parabolic comparison teorem yields te following corollary of Teorem 6.2. Corollary 6.3. For every α (0, 1 2 ), 0 < T < tere is a constant C = C(u 0,T,α) suc tat u u ɛ L (Ω (0,T )) Cɛ α. (6.11) We are now in position to look at te regularized level set mean curvature flow problem as a problem for graps. If we scale U = u ɛ (6.12) ɛ ten U is a solution of te mean curvature flow problem for graps (see (5.2)), tat is, U t 1+ U 2 U 1+ U 2 =0 inω (0,T). (6.13) Tis is a teoretical observation and implies tat we can apply tecniques developed for te mean curvature flow of graps to te mean curvature flow of level sets. But for computations we sall not use (6.13) but te unscaled version for u ɛ Te approximation of viscosity solutions Numerical scemes based on te level set approac were first introduced in Oser and Setian (1988); see also Setian (1990). Cen, Giga, Hitaka and Honma (1994) proposed a finite difference sceme for wic tey proved stability wit respect to te L -norm. Walkington (1996) used a finite element approac on te dual mes to construct a discretization tat is stable bot wit respect to L and to W 1,1. Evans (1993) analysed a sceme based on te solution of te usual eat equation, continually re-initialized after sort time-steps, and wic was proposed in Merriman, Bence and Oser (1994). Crandall and Lions (1996) constructed a finite difference sceme tat is bot monotone and consistent, and obtained te first convergence result for an approximation of (6.2), (6.3). An error analysis for tis sceme can be found in Deckelnick (2000). Here we want to consider a different finite element sceme wic exploits te above-described formal similarity to te grap case. Tis will also allow us to carry out some basic numerical analysis. In te following we use te abbreviations ( v, ɛ) ν ɛ (v) = Q ɛ (v), Q ɛ(v) = ɛ 2 + v 2, V ɛ (v) = v t Q ɛ (v). Our results for te mean curvature flow of a grap can directly be transformed into a convergence result for te regularized level set problem.

42 42 K. Deckelnick, G. Dziuk and C. M. Elliott Teorem 6.4. Let u ɛ be te solution of (6.8), (6.10) and let u ɛ be te solution of te semidiscrete problem u ɛ (,t) X wit u ɛ (,t) 1 X 0, u ɛ (, 0) = u 0 = I u 0 and u ɛt φ Q ɛ (u ɛ ) + u ɛ φ = 0 (6.14) Ω Q ɛ (u ɛ ) Ω for all t (0,T) and all discrete test functions φ X 0.Ten T (V ɛ (u ɛ ) V ɛ (u ɛ )) 2 Q ɛ (u ɛ ) c ɛ 2, 0 Ω Ω ν ɛ (u ɛ ) ν ɛ (u ɛ ) 2 Q ɛ (u ɛ ) c ɛ 2. Ω Ω sup (0,T ) We omit te proof as it is based on te scaling argument (6.12). Unfortunately, te constants c ɛ contain a term tat depends exponentially on 1 ɛ, wic is due to an application of Gronwall s lemma. Numerical tests, owever, suggest tat te resulting bound overestimates te error. In two space dimensions we can prove tat te computed solutions u ɛ converge in L to te viscosity solution. Te proof is contained in Deckelnick and Dziuk (2001). Teorem 6.5. Let u be te viscosity solution of (6.2), (6.3) and let u ɛ be te solution of te problem (6.14) wit Ω R 2 as in Corollary 6.3. Ten tere exists a function = (ɛ) 0asɛ 0 suc tat lim u u ɛ(ɛ) L ɛ 0 (Ω (0,T )) =0. Finally, te fully discrete numerical sceme for (regularized) isotropic mean curvature flow of level sets is now a straigtforward adaption of Algoritm 3. Algoritm 4. (Mean curvature flow of level sets) Let u 0 ɛ = I u 0. For m =0,...,M 1, compute u m+1 ɛ X suc tat u m+1 ɛ 1 X 0 and, for every φ X 0, m+1 m+1 1 uɛ φ τ Ω Q ɛ (u m ɛ ) + uɛ φ Ω Q ɛ (u m ɛ ) = 1 u m ɛ φ τ Ω Q ɛ (u m (6.15) ɛ ). For tis sceme we ave te following convergence result. Teorem 6.6. Let u ɛ be te solution of (6.8) (6.10) and let u m ɛ,(m = 1,...,M) be te solution from Algoritm 4. Ten tere exists a τ 0 > 0

43 Computation of geometric PDEs and mean curvature flow 43 suc tat, for all 0 <τ τ 0, M 1 τ (V ɛ (u m ɛ ) Vɛ m )2 Q ɛ (u m ɛ ) c ɛ(τ ), (6.16) m=0 Ω Ω sup ν ɛ (u m ɛ ) ν ɛ (u m ɛ ) 2 Q ɛ (u m ɛ ) c ɛ(τ ), (6.17) m=0,...,m Ω Ω wit M =[ T τ ]. Here V m ɛ = (um+1 ɛ u m ɛ )/(τ Q ɛ(u m ɛ )) is te regularized discrete normal velocity. Tis result implies te convergence of te fully discrete regularized solution to te viscosity solution. Teorem 6.7. Let u be te viscosity solution from Teorem 6.1 and let Ω be te domain from Corollary 6.3 in R 2. Let u ɛτ denote te timeinterpolated solution of te fully discrete sceme (6.15). Ten tere exist functions = (ɛ) 0andτ = τ(ɛ) 0asɛ 0 suc tat lim u u ɛ(ɛ)τ(ɛ) L ɛ 0 (Ω (0,T )) =0. 7. Pase field approac to mean curvature flow 7.1. Introduction Te pase field approac to interface evolution is based on pysical models for problems involving pase transitions. In tis section Ω is a bounded domain in R n+1 and Γ(t) is a ypersurface moving troug Ω. In te case of two pases one as te notion of an order parameter or pase field function ϕ :Ω (0,T) R wic indicates te pase of a material by associating wit te pases te minima of a C 2 double well bulk energy function W ( ) : R R. For simplicity we suppose tat W (r) =W ( r) and te minima of W ( ) areat ±1. Te canonical example is Consider te gradient energy functional E(ϕ) = W (r) = 1 4 (r2 1) 2. (7.1) Ω ( ɛ 2 ϕ 2 + W (ϕ) ɛ ) dx, (7.2) were ɛ is a small parameter. Steepest descent or gradient flow for tis functional leads to te parabolic Allen Can equation (Allen and Can 1979) ɛϕ t ɛ ϕ + 1 ɛ W (ϕ) =0 inω (0,T) (7.3)

44 44 K. Deckelnick, G. Dziuk and C. M. Elliott wit Neumann boundary conditions. In order to understand te beaviour of tis evolution equation for an initial function ϕ 0 :Ω R, observe tat te flow of te ordinary differential equation ϕ t = W (ϕ) drives positive ɛ 2 values of ϕ 0 to 1 and negative values to 1. On te oter and te Laplacian term in te equation (7.3) as a smooting effect wic will diffuse large gradients of te solution. Tus, on te basis of tese euristics, after a sort time te solution of (7.3) will develop a structure consisting of bulk regions in wic ϕ is smoot and takes te values ±1, and separating tese regions tere will be interfacial transition layers across wic ϕ canges rapidly from one bulk value to te oter. Tese transition layers are due to te interaction between te regularizing effect of te gradient energy term and te flow associated wit te bi-stable potential term W. It turns out tat te motion of tese interfacial transition layers approximate mean curvature flow. We can argue informally to support tis in te following way. Let, for t (0,T), Γ(t) be a smootly evolving closed and compact ypersurface satisfying V = H. Suppose tat Γ(t) is te boundary of an open set Ω(t) Ω and denote by d(,t) te signed distance function to Γ(t). We consider te semilinear parabolic operator P (v) =ɛv t ɛ v + 1 ɛ W (v). A calculation yields for v(x, t) =ψ ( d(x,t) ) ɛ, were ψ : R R, tat ( ) d P (v) =(d t d)ψ 1 ( ( ) ( ( ))) d d ψ W ψ. ɛ ɛ ɛ ɛ Hence it is natural to define ψ = ψ(z) to be te unique solution of ψ (z)+w (ψ(z)) = 0, z R, (7.4) ψ(z) ±1, z ±, ψ(0) = 0, ψ (z) > 0. (7.5) If W is given by (7.1) we ave tat ψ(z) = tan( z 2 ) and terefore ( ) d P (v) =(d t d)ψ. ɛ Recalling (2.12) and (2.20) we obtain d t d = V H =0onΓ(t), so tat te smootness of d implies d t d C d in a neigbourood U of 0<t<T Γ(t) {t}. Hence P (v) Cɛ d ( ) d ɛ ψ Cɛ in U ɛ and it follows tat v = ψ ( d ɛ ) is close to being a solution of (7.3) wit initial

45 Computation of geometric PDEs and mean curvature flow 45 data ϕ 0 = ψ ( d(,0) ) ɛ. Tat (7.3) is gradient flow for (7.2) is easily sown by testing te equation wit ϕ t and integrating by parts, leading to ɛ ϕ t 2 dx + de(ϕ) =0, Ω dt wic is te analogue of te energy equation in Lemma 3.1. A more general isotropic pase field equation is ɛϕ t = ɛ ϕ 1 ɛ W (ϕ)+c W g, (7.6) were g is a forcing term. Te constant c W is a scaling constant dependent on te precise definition of te double well potential W and is given by te formula c W = 1 1 W (r)dr. (7.7) 2 1 Te equation of motion tat tis pase field model approximates is V = H + g. (7.8) We refer to Rubinstein, Sternberg and Keller (1989) and de Mottoni and Scatzman (1995) for formal and rigorous interface asymptotics relating te Allen Can equation to mean curvature flow. Error bounds for te Hausdorff distance between te zero level set of te pase field function and te interface ave been derived (Cen 1992, Bellettini and Paolini 1996). In particular, a convergence rate of O(ε 2 log ε 2 ) was establised by Bellettini and Paolini (1996). Tese bounds are proved using comparison teorems for te pase field equation and tis can be extended to prove convergence to te viscosity solution of te level set equation in te case of nonsmoot evolution and witout te interface tickening (fattening) (Evans, Soner and Souganidis 1992) Te double obstacle pase field model We consider te pase field model ɛϕ t ε ϕ + 1 ɛ W (ϕ) =c W g. (7.9) Te potential W is taken to be of double obstacle form W (r) = 1 2 (1 r2 )+I [ 1,1] (r), (7.10) were { + for r > 1, I [ 1,1] (r) = (7.11) 0 for r 1, introduced in te gradient pase field models by Bellettini, Paolini and Verdi

46 46 K. Deckelnick, G. Dziuk and C. M. Elliott (1990), Blowey and Elliott (1991, 1993b), Cen and Elliott (1994), Paolini and Verdi (1992). Properly we sould interpret W (r) in te following way: (, 1] if r = 1, W (r) = r if r < 1, [ 1, ) if r =1. For tis potential, a calculation reveals tat te profile of te pase variable in te transition layer given by te solution of (7.4), (7.5) is 1 if r π 2, ψ(r) = sin(r) if r < π 2, 1 if r π 2. Furtermore, c W = π 4. Te double obstacle problem can be written in an equivalent variational inequality formulation. Let K be te convex set K = {η H 1 (Ω) : η 1}. Ten te problem is to seek ϕ L (0,T; K) H 1 (0,T; L 2 (Ω)) suc tat ϕ(, 0) = ϕ 0 and ε ϕ t (η ϕ)+ε ϕ (η ϕ) 1 ϕ(η ϕ) π g(η ϕ) (7.12) Ω Ω ε Ω 4 Ω for all η Kand for almost every t (0,T). It is well known tat tis problem as a unique solution. Teorem 7.1. Suppose tat te smoot ypersurfaces Γ(t) R n+1 satisfy: (i) Γ(t) = Ω(t) for open sets Ω(t) R n+1 ; (ii) tere exists δ>0 suc tat dist(γ(t), Ω) δ for t [0,T]; (iii) d t d D 0 d for d δ, t [0,T], were d(,t) is te signed distance function to Γ(t); (iv) V = H on Γ(t) fort [0,T]. Let ε be sufficiently small suc tat 1 2 πε δ(1 + 2e2D0T ) 1 and let ϕ = ϕ ε be te unique solution of (7.12) wit g = 0 and initial data ϕ 0 = ψ ( d(,0) ) ε. Ten, for all t [0,T], d(x, t) 1 2 πε(1 + 2e2D 0t ) ϕ ε (x, t) =1, d(x, t) 1 2 πε(1 + 2e2D 0t ) ϕ ε (x, t) = 1. Proof. See Cen and Elliott (1994).

47 Computation of geometric PDEs and mean curvature flow 47 A consequence of tis teorem is tat te diffuse interfacial region { (x, t) : ϕε (x, t) < 1 } is sarply defined wit finite widt bounded by c(t)πε and tat bot te zero level set of ϕ ε (,t)andγ(t) are in a narrow strip of widt c(t)πε. Here c(t) = 1 2 (1 + e2d0t ); but in practice it is observed tat tis is pessimistic and te growt of te interface widt is not usually an issue. A more refined analysis by Nocetto, Paolini and Verdi (1994) revealed in te case of a smoot evolution of te forced mean curvature flow tat te Hausdorff distance between te zero level set of ϕ ε and te interface of te flow (7.8) is of order O(ε 2 ). Furtermore, tere is convergence to te unique viscosity solution of te level set formulation (Nocetto and Verdi 1996a) Discretization of te Allen Can equation We use te same notation for te discrete spaces as in Section 5.2. We will identify any function Φ X wit te vector {Φ j } N j=1 of its nodal values, so tat Φ = N j=1 Φ jχ j.by(, ) wedenotetel 2 (Ω) inner product. For computational convenience we use a discrete inner product (, ) on C 0 (Ω) defined by (χ, η) = I (χη)dx for all χ, η C 0 (Ω), (7.13) Ω were I is te usual Lagrange interpolation operator for X. Furtermore, let τ = T/M > 0 be te uniform time-step and t m = mτ. For any {Φ m } M m=0, we set Φ m = τ 1 (Φ m+1 Φ m ). Te fully discrete approximation using explicit (θ = 0) and implicit (θ = 1) time-stepping reads as follows. Algoritm 5. (Allen Can equation) Let Φ 0 = I ϕ 0.Form =0,..., M 1, find Φ m+1 X,1 m M 1, suc tat, for all χ X, ( Φ m,χ) +( Φ m+θ, χ) 1 ε 2 (W (Φ m+θ ),χ) = c W ɛ (g, χ). (7.14) For initial data we coose te finite element interpolant of te transition layer profile ( ) Φ 0 = I d0 (x) ψ, ɛ were d 0 is te signed distance function to te initial interface. Te explicit sceme requires te usual time-step constraint for parabolic equations, τ C 2, (7.15) were te constant C depends on te mes and te L norm of te initial data troug te magnitude of W. On te oter and te implicit sceme

48 48 K. Deckelnick, G. Dziuk and C. M. Elliott requires a time-step constraint in order for te nonlinear equations defining Φ m+1 to ave a unique solution. Tis constraint is τ αɛ 2, (7.16) were α is te minimum value of W. See Elliott and Stuart (1993) and Cen, Elliott, Gardiner and Zao (1998). Te analysis of convergence to mean curvature flow requires consideration of te tree approximation parameters ɛ,, τ tending to zero. Standard apriorifinite element error analysis for fixed ɛ would lead to, for te difference between te finite element solution and te solution of te Allen Can equation, optimal order error bounds in terms of te mes sizes τ, but wit constants depending on te Gronwall-induced factor exp( T ). Feng ɛ 2 and Prol (2003) ave improved te finite element error analysis of te Allen Can equation using te special structure of te solution. Indeed, tey exploit spectral estimates of Cen (1994) wic lead to error bounds wose constants sow just polynomial growt in 1 ɛ. Tey specifically consider te implicit sceme witout numerical integration. As a consequence tey derive an error bound of order ɛ 2 between te zero level set of te solution of te Allen Can equation and te limiting surface Discretization of te double obstacle pase field model We use te finite element setting of Section 7.3. Let K = {χ X : χ 1}. Te double obstacle version of Algoritm 5 is as follows. Algoritm 6. (Double obstacle pase field) Let Φ 0 = I ϕ 0.Form = 0,...,M 1, find Φ m+n K suc tat, for all χ K, ( Φ m,χ Φ m+1 ) +( Φ m+θ, χ Φ m+1 ) (7.17) 1 ε 2 (Φm+θ + ɛc W g m+θ,χ Φ m+1 ) 0. For initial data we coose te finite element interpolant of te transition layer profile. Te explicit sceme is a discrete obstacle variational inequality associated wit te mass matrix. Witout mass lumping te solution of tis nonlinear algebraic problem would require quadratic programming or linear complementarity metods. However, wit te mass lumping quadrature rule te explicit sceme is as simple as te explicit sceme for a semilinear parabolic equation. It can be simply written as Φ m+1/2 = ((1+ τɛ ) ) 2 I τa Φ m + c W τ ɛ gm, (7.18) Φ m+1 = PΦ m+1/2. (7.19) Here A = M 1 K, were M and K are defined by M ij =(χ i,χ j ), K ij =( χ i, χ j ),

49 Computation of geometric PDEs and mean curvature flow 49 for 1 i, j N. Furtermore, P : R N R N is te component-wise projection onto [ 1, 1] N defined by (PV ) j = max( 1, min(1,v j )). On te oter and, in linear algebraic form te implicit sceme leads to te discrete variational inequality: find Φ m+1 R N suc tat Φ j 1and ((1 τ ) ) ɛ 2 I+τA Φ m+1 (χ Φ m+1) ( Φ m τ +c gm+1) (χ Φ m+1) W (7.20) ɛ for all χ R N wit χ j 1. Because A is symmetric tis is equivalent to minimizing a quadratic function subject to bound constraints and can easily be solved by projected SOR (Elliott and Ockendon 1982). Suc a system can also be solved by nonlinear multigrid (Kornuber and Krause 2003). As for te continuous parabolic variational inequality, a discrete comparison principle olds for tese scemes if te triangulation is acute. Tis provides te basis for a convergence analysis (Nocetto and Verdi 1996b, 1997). For te implicit sceme witout numerical integration an O(ɛ) error bound for te interface is obtained wen τ = O( 2 )=O(ɛ 4 ). For te explicit sceme witout numerical integration in te potential term an O(ɛ 2 )is proved for τ = O( 2 )=O(ɛ 5 ) Implementation One expects tere to be a relationsip between ɛ and in order tat te discrete pase field model can approximate te sarp interface motion. Since te convergence analysis in te continuous case relies eavily on understanding te profile of te pase field function across te transition layer, one would expect tat for any ɛ te mes size sould be sufficiently small in order to resolve te interface. Indeed te existing convergence analysis described above indicates tat sould tend to zero faster tan ɛ. In practice tis implies tat across te discrete interfacial layer in te normal direction tere sould be a sufficient number of elements. In te case of te double obstacle potential, at te mt time-step, te finite elements may be divided into tree sets: J (m) ={Φ j = 1 for eac element vertex}, J+(m) ={Φ j = 1 for eac element vertex}, I (m) =T \ (J+(m) J+(m)). Clearly te approximation to te interface is te zero level set of Φ m wic lies inside te discrete interfacial region I (m). We view I (m) asasarp diffuse interface, as opposed to te interfacial region associated wit te smoot double well, wic is not sarply defined. Te computational work in evolving te interface is ten associated wit te small number of

50 50 K. Deckelnick, G. Dziuk and C. M. Elliott Figure 7.1. Meses. elements in tis region. As observed above, te time-step, τ, in te pase field calculations is substantially smaller tan te mes size,. Tus, in a numerical simulation one would expect tat, for finite normal velocity of te interface, te sarp diffuse interface sould only move by at most te addition or subtraction of a single layer of elements. In te case of te explicit sceme tis can be made precise. For nodes in J+(m) (orj (m)) wose nearest neigbours are also in J+(m) (orj (m)), we find Φ m+1/2 j = ±1+ τ ɛ 2 (±1+c W ɛg m (a j )) wic, provided g m (a j ) 1 c W ɛ, implies tat Φm+1 j = ±1. It follows tat te sarp diffuse interface can not move more tan one element per timestep. It also implies tat it is only necessary to compute Φ m+1 on te closure of te transition layer. Tis can be exploited in a number of ways. Te two-dimensional dynamic mes algoritm (Nocetto, Paolini and Verdi 1996) is based on te explicit sceme and carries a mes only in te sarp diffuse interface; it adds and removes triangles were necessary. Te mask metod (Elliott and Gardiner 1996) keeps an underlying fixed mes and computes in te sarp diffuse interface only. It is possible to store nodal values only in tis region. An amalgam of te above is an adaptive procedure wic uses a fine mes witin te diffuse interface and a coarse mes outside. In Figure 7.1 a typical mes is sown for a pase field calculation of anisotropic mean curvature flow. Te global mes is sown togeter wit a zoom. Tis approac requires a fine mes sligtly larger tan te diffuse interface. As te interface region moves te mes is refined and coarsened appropriately.

51 Computation of geometric PDEs and mean curvature flow Figure 7.2. Topological cange Figure 7.3. Diffuse interfaces wit topological cange. Sarp diffuse interface front tracking Using te double obstacle pase field metod and only computing witin a sarp diffuse interface as described above can be viewed as a front tracking metod, wic as te advantage of being able to andle topological cange. In Figure 7.2 te interfaces at various times are displayed for a forced mean curvature flow starting from initial circles. Eventually te circles intersect. Meses associated wit tese computations are sown in Figure Anisotropic mean curvature flow 8.1. Te concept of anisotropy In free boundary problems suc as pase transition problems it is often necessary to treat interfaces wic are driven by anisotropic curvature. Tis is induced by modelling an anisotropic surface energy, wic generalizes area in te isotropic case to weigted area in te anisotropic case. Anisotropic

52 52 K. Deckelnick, G. Dziuk and C. M. Elliott surface energy as te form E γ (Γ) = Γ γ(ν)da, (8.1) were Γ is a surface wit normal ν and γ is a given anisotropy function. For γ(p) = p tis energy is te area of Γ. For our purposes it will be necessary to restrict te admissible anisotropies to a certain class. Definition 4. An anisotropy function γ : R n+1 R is called admissible if (1) γ C 3 (R n+1 \{0}), γ(p) > 0forp R n+1 \{0}; (2) γ is positively omogeneous of degree one, i.e., (3) tere exists γ 0 > 0 suc tat γ(λp) = λ γ(p) for all λ 0,p 0; (8.2) D 2 γ(p)q q γ 0 q 2 for all p, q R n+1, p =1, p q =0. (8.3) It is not difficult to verify tat (8.2) implies Dγ(p) p = γ(p), D 2 γ(p)p q =0, (8.4) Dγ(λp) = λ λ Dγ(p), D2 γ(λp) = 1 λ D2 γ(p) (8.5) for all p R n+1 \{0}, q R n+1 and λ 0. Te convexity assumption (8.3) will be crucial for analysis and numerical metods. Anisotropy is normally visualized by using te Frank diagram F and te Wulff sape W: F = {p R n+1 γ(p) 1}, W = {q R n+1 γ (q) 1}. Figure 8.1. Frank diagram (left) and Wulff sape (rigt) for te regularized l 1 -anisotropy γ(p) = 3 j=1 ε 2 p 2 + p 2 j.

53 Computation of geometric PDEs and mean curvature flow 53 Here γ is te dual of γ, wic is given by γ (q) = p q sup p R n+1 \{0} γ(p). (8.6) Let us consider some examples. Note tat not all of tem are admissible. Te coice γ(p) = p is called te isotropic case; in particular we ave tat F = W = {p R n+1 p 1} is te closed unit ball. A typical coice for anisotropy is te discrete l r -norm for 1 r, γ(p) = p l r = ( n+1 k=1 p k r ) 1 r, 1 r<, (8.7) wit te obvious modification for r =. For a given positive definite (n + 1) (n + 1) matrix G, te anisotropy function γ(p) = Gp p (8.8) models an anisotropy wic is defined by a (constant) Riemannian metric. In Figure 8.2 we sow te Frank diagram and Wulff sape for te anisotropy γ(p) = ( sign(p 1 ))p p2 2 + p2 3. (8.9) One anisotropy function often used in a pysical context is ( ( )) γ(p) = 1 A 1 p 4 l 4 p 4 p l 2 (8.10) l 2 were A is a parameter. For A<0.25 te Frank diagram is convex. For more information on tis subject, including anisotropies tat may depend on space, see Bellettini and Paolini (1996). Figure 8.2. Frank diagram F (left) and Wulff sape W (rigt) for te anisotropy (8.9).

54 54 K. Deckelnick, G. Dziuk and C. M. Elliott 8.2. Anisotropic distance function Let γ be an admissible anisotropy function. nonsymmetric metric Υ on R n+1 by setting We can associate wit γ a Υ(x, y) =γ (x y), x,y R n+1. (8.11) It is possible to prove tat Υ is equivalent to te standard Euclidean metric. Suppose next tat Ω R n+1 is a bounded open set wit smoot boundary Γ. Using Υ we now define an anisotropic signed distance function d γ : R n+1 R by inf y Γ Υ(x, y), x R n+1 \ Ω, d γ (x) = 0, x Γ, inf y Γ Υ(x, y), x Ω. Lemma 8.1. C 2 (U) and Tere exists an open neigbourood U of Γ suc tat d γ 8.3. Anisotropic mean curvature γ( d γ )=1, (8.12) D 2 d γ Dγ( d γ )=0. (8.13) Our goal is to generalize te notion of mean curvature to te anisotropic setting. Suppose tat γ is an admissible anisotropy function and tat Γ R n+1 is an oriented ypersurface wit normal ν. We define te Can Hoffmann vector ν γ on Γ by and te anisotropic mean curvature by ν γ (x) =Dγ(ν(x)), x Γ, (8.14) H γ (x) = Γ ν γ (x), x Γ. (8.15) Note tat H γ = H in te isotropic case γ(p) = p. Te following lemma sows tat H γ is a natural generalization of mean curvature as te first variation of te area functional wit respect to normal variations. Lemma 8.2. Suppose tat Γ is compact. For φ C0 (U) (U a neigbourood of Γ) define F ɛ (x) =x + ɛφ(x)ν(x),x U as well as Γ ɛ = F ɛ (Γ). Ten, d dɛ E γ(γ ɛ ) ɛ=0 = H γ φ da. Γ Proof. Let d(,ɛ):r n+1 R denote te signed distance function to Γ ɛ. Consider g : U ( ɛ 0,ɛ 0 ) R, defined by g(x, ɛ) =γ(ν ɛ (x)) = γ( d(x, ɛ)),

55 Computation of geometric PDEs and mean curvature flow 55 were acts on te x variables only. Now (2.23), (2.20) and (2.6) imply d dɛ E γ(γ ɛ ) ɛ=0 = d g(,ɛ)da dɛ ɛ=0 Γ ɛ g = (, 0) da g(, 0) d g (, 0)H da (, 0) d(, 0) da. Γ ɛ Γ ɛ Γ ν ɛ It is not difficult to see tat d ɛ (, 0) = φ(x),x Γ, wic also implies tat g ɛ (, 0) = γ (ν) d ɛ (, 0) = d γ (ν) Γ ɛ (, 0) = ν γ Γ φ. Here we ave used te definition of ν γ and te fact tat d (, 0) ν =0on Γ. Tus, d dɛ E γ(γ ɛ ) ɛ=0 = ν γ Γ φ da + Γ = Γ ν γ φ da + Γ Γ Γ γ(ν)φh da + Γ g (, 0)φ da, ν ɛ g (, 0)φ da ν were te last identity follows from (2.16). Finally, observing tat g ν (, 0) = γ pi (ν)d xi x j (, 0)d xj (, 0) = 0, and recalling te definition of H γ, te claim follows. Let us next calculate H γ for various descriptions of Γ. Level set representation. Suppose tat Γ is given as in (2.1) and oriented by ν = u u. Since γ p i is omogeneous of degree 0, we ave (see also (2.2)) n+1 ( ( )) n+1 u ( H γ = Γ ν γ = D i γ pi = D u i γpi ( u) ) = n+1 i,j=1 i=1 γ pi p j ( u)u xi x j Recalling (8.4) we terefore deduce H γ = n+1 i,j=1 n+1 i,k,l=1 i=1 u xk u xi γ pi p l ( u)u xl x k u u. γ pi p j ( u)u xi x j. (8.16) Grap representation. If Γ is locally given as te grap of te function x v(x ),x =(x 1,...,x n ) wit normal ν = ( x v, 1), formula (8.16) 1+ x v 2 applied to u(x,x n+1 )=v(x ) x n+1 gives n H γ = γ pi p j ( x v, 1)v xi x j. (8.17) i,j=1

56 56 K. Deckelnick, G. Dziuk and C. M. Elliott Let us next derive an analogue of (2.16) wit H replaced by H γ. Observing tat D k ν l = D l ν k and recalling tat D k x l = δ kl ν k ν l, we obtain H γ ν l = D k ( γpk (ν) ) ν l = D k ( γpk (ν)ν l ) γpk (ν)d k ν l ( ) = D k γpk (ν)ν l γpk (ν)d l ν k ( ) ( ) = D k γpk (ν)ν l Dl γ(ν) ( ) ( = D k γpk (ν)ν l Dk γ(ν)(δkl ν k ν l ) ) γ(ν)ν l D k ν k = D k ( γpk (ν)ν l ) Dk ( γ(ν)dk x l ) γ(ν)hνl, were summation over k is from 1 to n + 1. For a smoot test function φ =(φ 1,...,φ n+1 ) we multiply te above relation by φ l,sumoverl and integrate over Γ. Using (2.16) we infer Γ H γ ν φ = and (8.4) yields H γ ν φ = Γ n+1 k,l=1 + n+1 k,l=1 n+1 k,l=1 Γ Γ γ pk (ν)ν l D k φ l + Γ n+1 k,l=1 n+1 γ(ν)d k x l D k φ l γ pk (ν)ν l D k φ l + n+1 k,l=1 Γ Γ l=1 γ pk (ν)ν l Hν k φ l Γ γ(ν)hν l φ l γ(ν)d k x l D k φ l. (8.18) Tis relation will be at te eart of te numerical metods in te parametric case. For additional information on te subject of weigted mean curvature including te crystalline case, see Taylor (1992) Motion by anisotropic mean curvature wit mobility Having introduced te notion of anisotropic mean curvature we can now formulate te following generalization of (3.1): β(ν)v = H γ + g on Γ(t). (8.19) Here, β : S n R is a given positive and smoot function of degree zero. In applications were Γ(t) models a sarp pase-interface, te coefficient β measures te drag opposing interfacial motion and te function 1 β is called mobility. Te function g represents te energy difference in te bulk pases. A detailed derivation of (8.19) from te force balances and te second law of termodynamics can be found in Angenent and Gurtin (1989) and Gurtin (1993). Taylor, Can and Handwerker (1992) give an overview of various matematical approaces to (8.19).

57 Computation of geometric PDEs and mean curvature flow 57 In wat follows we sall consider te simpler problem β(ν)v = H γ on Γ(t), (8.20) even toug all our tecniques can be generalized to (8.19). It can be sown (see Bellettini and Paolini (1996)) tat for te coice β(ν) = 1 γ(ν) tere is an explicit solution of (8.20) consisting of srinking boundaries of Wulff sapes; te sets Γ(t) ={p R n+1 γ (p) = r(0) 2 2nt} satisfy 1 γ(ν) V = H γ and are terefore a generalization of te srinking circles from te isotropic case. We also ave te following analogue of Lemma 3.1. Lemma 8.3. Let Γ(t) be a family of evolving ypersurfaces satisfying (8.20) on Γ(t), and assume tat eac Γ(t) is compact. Ten β(ν)v 2 da + d γ(ν) =0. Γ(t) dt Γ(t) Proof. In te same way as in te proof of Lemma 8.2, we derive d γ(ν) = H γ V, dt Γ(t) Γ(t) and te claim follows from te evolution law (8.20) Anisotropic curve sortening flow Let us consider a family Γ(t) ofclosedcurvesinr 2 wic move according to (8.20). As in Section 4.1 we describe te evolution by means of a mapping X : R [0,T) R 2 wic satisfies X(θ, t) =X(θ +2π, t) fort [0,T), θ R. Te curves Γ(t) = X(,t) will move by (8.20) provided tat β(ν)x t = H γ ν. (8.21) Using te notation (a 1,a 2 ) = ( a 2,a 1 ) we may write ν = τ, were τ = X θ X θ is te unit tangent to te curve Γ(t). Equation (8.21) amounts to a system of partial differential equations for te vector function X. In order to write down tis system, let ϕ Hper(I; 1 R 2 ),I=[0, 2π], be a test function, wic we can tink of as being defined on Γ(t) via φ(x(θ, t)) = ϕ(θ). It follows from (8.18) tat 2 2 H γ ν φ = γ pk (ν)ν l D k φ l + γ(ν)d k x l D k φ l Γ(t) = k,l=1 2 k,l=1 Γ(t) Γ(t) k,l=1 Γ(t) ( γpk (ν)ν l γ(ν)δ kl ) Dk φ l,

58 58 K. Deckelnick, G. Dziuk and C. M. Elliott since D k x l = δ kl ν k ν l. Using Γ φ l = ϕ l,θ X θ τ and recalling tat γ(p) = Dγ(p) p, we obtain after some calculations 2 ( ) ϕ θ γpk (ν)ν l γ(ν)δ kl Dk φ l = Dγ(ν) X θ. k,l=1 In conclusion we ave H γ ν ϕ da =+ 0 β( X θ X θ Γ(t) 2π 0 Dγ(X θ ) ϕ θ dθ, so tat we obtain te following weak form of (8.21): 2π ) 2π X t ϕ X θ + 0 Dγ(X θ ) ϕ θ dθ =0 forallϕ H1 per(i; R 2 ). (8.22) We sall base our numerical sceme on tis formulation. Te classical form of (8.22) is X β( θ X θ ) X t + 1 X β( θ ( ( ) Dγ X ) X θ X θ θ θ =0 ini (0,T). (8.23) For te convenience of te reader we explicitly write down te two equations of tis system: ) X 1t X θ γ p2 p 2 ( X 2θ,X 1θ )X 1θθ + γ p2 p 1 ( X 2θ,X 1θ )X 2θθ =0, β( X θ X θ ) X 2t X θ γ p1 p 1 ( X 2θ,X 1θ )X 2θθ + γ p1 p 2 ( X 2θ,X 1θ )X 1θθ =0. It is easy to see tat tis system can be written as X ) β( θ X ) ( ) X t a( θ 1 Xθ =0, X θ X θ X θ θ X θ were a(p) =γ pp (p) p p, p R 2 \{0}. Note tat (8.3) implies a(p) γ 0 > 0 for all p R 2, p = 1. Analytical results for tis problem wic generalize te teory for te isotropic case (a = 1) ave been obtained by Gage (1993). We sall continue to use te form (8.22) because tis equation only contains first derivatives of te anisotropy function γ. Recall te definition of S from Section 4.1. A discrete solution of (8.22) will be a function X :[0,T] S, suc tat X (, 0) = X 0 = I X 0 = N X 0 (θ j )φ j, j=1

59 Computation of geometric PDEs and mean curvature flow 59 and for all discrete test functions ϕ S 2π ( ) 2π Xθ β X t ϕ X θ dθ + Dγ(Xθ 0 X θ ) ϕ θ 0 In te same way as in te isotropic case we can write X (θ, t) = N X j (t)φ j (θ) j=1 dθ =0. (8.24) wit X j (t) R 2, and find tat te discrete weak equation (8.24) is equivalent to te following system of 2N ordinary differential equations: 1 6 β jq j Ẋ j (β jq j + β j+1 q j+1 )Ẋj β j+1q j+1 Ẋ j+1 + Dγ(X j+1 X j ) Dγ(X j X j 1) =0, for j =1,...,N, were X 0 (t) =X N (t), X N+1 = X 1 (t), and ( ) Xj X j 1 q j = X j X j 1, β j = β. Furtermore, te initial values are given by q j X j (0) = X 0 (θ j ), j =1,...,N. We again use mass lumping, wic is equivalent to a quadrature formula. Tus we replace tis system by te lumped sceme 1 2 (β jq j + β j+1 q j+1 )Ẋj + Dγ(Xj+1 Xj ) Dγ(Xj Xj 1) = 0 (8.25) togeter wit te initial conditions X j (0) = X 0 (θ j )forj =1,...,N. We are now ready to say wat we mean by a discrete solution of anisotropic curve sortening flow. Te above system is equivalent to te one we use in te following definition of discrete anisotropic curve sortening flow. Definition 5. A solution of te discrete anisotropic curve sortening flow for te initial curve Γ 0 = X 0 ([0, 2π]) is a polygon Γ (t) =X ([0, 2π],t), wic is parametrized by a piecewise linear mapping X (,t) S, t [0,T], suc tat X (, 0) = X 0 and for all ϕ S 2π ( ) 2π Xθ β X t ϕ X θ dθ + Dγ(Xθ 0 X θ ) ϕ θ dθ π ( ) Xθ 6 2 β X θt ϕ θ X θ dθ =0. (8.26) X θ 0

60 60 K. Deckelnick, G. Dziuk and C. M. Elliott Figure 8.3. Anisotropic curve sortening flow wit a sixfold anisotropy function applied to a circle (left) and to a square (rigt). Here is te constant grid size of te uniform grid in [0, 2π]. Te last term of (8.26) is introduced by mass lumping. One could also define te discrete curve sortening flow witout tis quantity, but ten te geometric property of lengt sortening would not be true for te discrete problem. Dziuk (1999b) proved te following convergence result for β = 1. It is easily extended to te case of general β. We formulate te result for te geometric quantities normal, lengt and normal velocity. Te error estimates in standard norms ten follow easily. Teorem 8.4. Suppose tat β : S 2 R is a smoot positive function. Let X be a solution of te anisotropic curve sortening flow (8.23) on te interval [0,T]witX(, 0) = X 0,min [0,2π] [0,T ] X θ c 0 > 0andX t L 2 ((0,T),H 2 (0, 2π)). Ten tere is an 0 > 0 suc tat, for all 0 < 0, tere exists a unique solution X of te discrete anisotropic curve sortening flow (8.26) on [0,T]witX (, 0) = X 0 = I X 0, and te error between smoot and discrete solution can be estimated as follows: sup (0,T ) 2π 0 ν ν 2 X θ dθ +sup (0,T ) T 2π 0 0 2π 0 ( X θ X θ ) 2 dθ c 2, X t X t 2 X θ dθ dt c 2. Te constants depend on c 0, T and X t L 2 ((0,T ),H 2 (0,2π)).

61 Computation of geometric PDEs and mean curvature flow 61 Table 8.1. Convergence test for anisotropic curve sortening flow. E 1 EOC 1 E 2 EOC 2 E 3 EOC 3 E 4 EOC We tested te algoritm wit an exact solution, X(θ, t) = 1 2t (cos g(θ), sin g(θ)), were we ave cosen g(θ) = θ + 0.1sinθ. γ(p) = p 0.25p 1. We compute te errors Te anisotropy function is E 1 = X t X t L 2 ((0,T ),L 2 (Γ )), E 2 = ν ν L ((0,T ),L 2 (Γ )), E 3 = X θ X θ L 2 ((0,T ),L 2 (S 1 )), E 4 = X θ X θ L ((0,T ),L 2 (S 1 )) wit Γ = X ((0, 2π), ). For two successive grid sizes 1 and 2 wit corresponding errors E( 1 )ande( 2 ), te experimental order of convergence EOC = ln (E( 1 )/E( 2 ))/ ln ( 1 / 2 ) is calculated and sown in Table 8.1 from Dziuk (1999b). Te time-step τ was cosen τ =0.5 2 for tese computations. We empasize tat te algoritm for anisotropic curve sortening flow does not use te second derivatives of te anisotropy function γ. Te system (8.25) can be formally written in complex tridiagonal form. For details and a suitable time discretization we refer to Dziuk (1999b). Let us finally mention tat in Girao (1995) simple closed convex curves evolving by (8.20) are computed by approximating te smoot anisotropy by a crystalline one. Also, an error analysis for te resulting metod is given Anisotropic curvature flow of graps Let us next turn to te evolution of ypersurfaces wic are given as graps, i.e., Γ(t) = {(x, u(x, t)) x Ω}. In order to translate (8.20) into an evolution equation for u we recall tat n n ( H γ = γ pi p j ( u, 1)u xi x j = γpi ( u, 1) ). x i i,j=1 i=1

62 62 K. Deckelnick, G. Dziuk and C. M. Elliott Furtermore, since V = ut Q wit Q = 1+ u 2 we see tat (8.20) leads to te following nonlinear partial differential equation, ( ( u, 1) n β Q ) u t Q i=1 x i (γ pi ( u, 1)) = 0 in Ω (0,T), (8.27) to wic we add te following initial and boundary conditions: u = g on Ω (0,T), u(, 0) = u 0 in Ω. In te sequel we sall again assume tat tis problem as a solution u wic satisfies (5.12) and refer to Deckelnick and Dziuk (1999) for a corresponding existence and uniqueness result. Discretization in space and estimate of te error As in te isotropic case we may use a variational approac even toug te differential equation is not in divergence form. Starting from (8.27) we obtain, wit te abbreviation ν = ( u, 1) Ω β(ν)u t ϕ Q + n i=1 Ω Q, γ pi ( u, 1)ϕ xi = 0 (8.28) for all ϕ H0 1 (Ω), t (0,T) togeter wit te above initial and boundary conditions. We now consider a semidiscrete approximation of (8.28): find u (,t) X wit u (,t) I g X 0 suc tat β(ν )u,t ϕ n + γ pi ( u, 1)ϕ,xi =0 forallϕ X 0, (8.29) Ω Q i=1 Ω for all t (0,T], were we ave set Q = 1+ u 2, ν = ( u, 1). Q As an initial condition we use u (, 0) = u 0 = I u 0. Our main result gives an error bound for te important geometric quantities V and ν. Te proof is contained in Deckelnick and Dziuk (1999). Teorem 8.5. Suppose tat (8.27) as a solution u tat satisfies (5.12). Ten (8.29) as a unique solution u and T 0 V V 2 L 2 (Γ (t)) dt + sup t (0,T ) (ν ν )(,t) 2 L 2 (Γ (t)) C2. Here, Γ (t) ={(x, u (x, t)) x Ω Ω} and V, V are as in Teorem 5.4.

63 Computation of geometric PDEs and mean curvature flow 63 Fully discrete sceme, stability and error estimate Let us next consider discretization in time in order to get a practical metod. Compared to te isotropic case, our problem as become more complicated because of te presence of two additional nonlinearities, namely te functions β and γ. In order to keep te computational effort as small as possible it would be desirable to ave a metod tat only requires te solution of a linear problem at eac time-step. Tis can be acieved by treating te nonlinearities in an explicit way and guaranteeing stability via te introduction of an additional stabilizing term. We start again from te variational formulation (8.28) and coose a time-step τ>0. Using te notation from Section 5.3 our sceme reads as follows. Algoritm 7. (Anisotropic mean curvature flow of graps) Given u m, find um+1 X suc tat u m+1 I g X 0 and 1 β(ν m)(u m+1 u m ) n ϕ + γ pi (ν m τ )ϕ x i Ω Ω + λ Q m Ω γ(ν m ) Q m i=1 (u m+1 u m ) ϕ = 0 (8.30) for all ϕ X 0. Here we ave set u 0 = I u 0 as well as = 1+ u m 2, ν m = ( um, 1). Q m Te above sceme is semi-implicit and requires te solution of a linear system in eac time-step. We sall see tat it is unconditionally stable provided te parameter λ is cosen appropriately. 1 Teorem 8.6. Let γ = 5 1 max { sup p =1 γ(p), sup p =1 D 2 γ(p) }. Ten we ave for 0 M [ T τ ] M 1 β(ν m τ ) u m+1 u m 2 M 1 γ(ν m m=1 Ω Q m τ + λτ ) ( Q m+1 Q m ) 2 m=1 Ω Q m τ ( ) M 1 + λ inf γ(p) γ τ ν m+1 ν m p =1 2Q m+1 m=1 Ω τ + γ(ν M )QM Ω γ(ν 0 )Q0. Ω In particular, if λ iscoseninsucawaytatλ inf p =1 γ(p) > γ, ten we ave for Γ m = {(x, um (x)) x Ω } [ ] T E γ (Γ m ) E γ(γ 0 ) for all 0 m. (8.31) τ Q m

64 64 K. Deckelnick, G. Dziuk and C. M. Elliott Tus we ave proved stability for te semi-implicit sceme witout any restriction on te time-step size. An error analysis for te above sceme as been carried out in Deckelnick and Dziuk (2002a). Te precise result is as follows. Teorem 8.7. Suppose tat λ inf p =1 γ(p) > γ (γ as in Teorem 8.6). Ten tere exists τ 0 > 0 suc tat, for all 0 <τ τ 0, [ T τ ] 1 m=0 τ (V m V m )2 Q m + Ω Ω max 0 m [ T τ ] Ω Ω ν m ν m 2 Q m c(τ ). We ave run numerical tests for anisotropic mean curvature flow of graps. Te Wulff sape srinks omotetically during te evolution. We ave cosen te very strong anisotropy γ(p) = 0.01p p2 2 + p2 3. Te equation γ (x, u(x, t)) = 1 4t defines a solution of te differential equation wen te mobility is cosen as β =1/γ. Te exact solution is given by u(x, t) = 1 4t 100x 2 1 x2 2. Te condition on te stabilizing parameter λ (see Teorem 8.6) is satisfied for λ =81.0. We use τ =0.01 as a uniform time-step size. Te coupling between time-step size and spatial grid size is done in order not to spoil te asymptotic orders of convergence. For a discussion wit respect to te coice of λ and τ we refer to Deckelnick and Dziuk (2002a). Table 8.2 sows te grid size, te errors ( M E(V )= τ V m V m 2 Q m m=0 Ω ( E(ν) = max ν m ν m 0 m M 2 Q m Ω ) 1 2, ) 1 2, Table 8.2. Convergence test for anisotropic mean curvature flow of graps. E(ν) EOC E(V ) EOC L (H 1 ) EOC e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e

65 Computation of geometric PDEs and mean curvature flow 65 Figure 8.4. Level lines for te time-steps 0, 250, 500, 750, 3000 for a regularized crystalline anisotropy. Figure 8.5. Initial value and stationary solution for a regularized crystalline anisotropy. and te corresponding experimental orders of convergence (EOC) between two successive grid sizes. We add a column wit te L ((0,T),H 1 (Ω)) error. Obviously te results of te asymptotic error estimates of Teorem 8.7 are reproduced in our test computations. We add a long time computation from Deckelnick and Dziuk (2002a). As an anisotropy function we used an anisotropy wic is a regularized form of γ(p) = p l (see also Figure 8.1 for te dual situation). We ave cosen a nonzero constant rigt-and side for te equation. In Figure 8.4 we sow te level lines of te initial function, of four time-steps and of te stationary solution. Te boundary values were kept fixed during te evolution. We can see tat te octaedral sape develops during te evolution. In Figure 8.5 we sow te initial grap and te stationary grap. Te domain was te unit disk. Figure 8.4 sows te evolution of some levels of te grap.

66 66 K. Deckelnick, G. Dziuk and C. M. Elliott 8.7. Anisotropic mean curvature flow of level sets Here we briefly sketc ow te level set approac can be adapted to te anisotropic case. Let us look for solutions of (8.20) in te form Γ(t) ={x R n+1 u(x, t) =0} were te scalar function u as to be determined. Te relations (2.20) and (8.16) lead to te following nonlinear partial differential equation ( ) u n β u t u γ pj p u k ( u)u xj x k =0 inr n+1 (0, ), (8.32) j,k=1 wic is degenerate parabolic since D 2 γ(p)p = 0. We regularize te equation by using an extension of te anisotropy to n + 2 space dimensions. Let us assume tat tere is an admissible weigt function γ = γ(p 1,...,p n+1,p n+2 ) suc tat γ(p 1,...,p n+1, 0) = γ(p 1,...,p n+1 ). In te following we denote tis extension again by γ. Rater tan treating (8.32) we introduce for a (small) positive parameter ɛ te regularized problem ( ) u ɛ β u ɛt n+1 ɛ 2 + u ɛ 2 γ ɛ 2 + u ɛ 2 pj p k ( u ɛ, ɛ)u ɛxj x k =0. j,k=1 We consider tis differential equation on Ω (0,T), were Ω R n+1 is a bounded smoot domain and T > 0 is some final time. Furtermore, appropriate initial and boundary conditions need to be added, wic can be done similarly to te isotropic case. Te numerical approximation of te resulting problem follows te ideas of te grap case. Te same applies to te analysis of te scemes, were of course one as to bear in mind te dependency on te regularization parameter ɛ Anisotropic pase field We turn now to te setting of Section 7. Anisotropic pase field models are based on te following anisotropic interfacial energy functional Eγ(ϕ) ɛ = (ɛa( ϕ)+ 1ɛ ) W (ϕ) dx, (8.33) Ω were A : R n+1 R is smoot, convex and positively omogeneous of degree two wic replaces te quadratic gradient energy used in te isotropic case. In order to relate it to te anisotropic energy density used in tis section we set A(p) = 1 2 γ(p)2, p R n+1. (8.34)

67 Computation of geometric PDEs and mean curvature flow 67 Te double well bulk energy function W may be cosen as in te isotropic situation. Te L 2 -gradient flow of E γ leads to te following quasilinear parabolic equation: ɛϕ t ɛ DA( ϕ)+ 1 ɛ W (ϕ) =0 inω (0,T). (8.35) For small ɛ and suitable initial data, (8.35) approximates te following anisotropic mean curvature flow: 1 γ(ν) V = H γ on Γ(t). (8.36) Tis can be motivated in a similar manner to te isotropic case. For convenience we suppose tat W is a smoot double well. Set P (v) =ɛv t ɛ DA( v)+ 1 ɛ W (v) (8.37) and ( ) dγ (x, t) v(x, t) =ψ, ɛ were ψ is te transition profile defined by (7.4) and (7.5) and d γ (,t)denotes te anisotropic signed distance function to te smootly evolving interface Γ(t) wic satisfies (8.36) on Γ(t). A calculation sows ( ) ( ) v t = ψ dγ dγ,t dγ dγ, v = ψ, D 2 v = ψ ɛ ( dγ ɛ ɛ ) dγ d γ ɛ 2 + ψ ( dγ ɛ ɛ ɛ ) D 2 d γ, ɛ and using (8.4), (8.5) as well as Lemma 8.1, we obtain ( ) ( ( ) n+1 ) P (v) =ψ dγ d γ,t d γ ɛ d γ γ dγ γ pi p d γ j ( d γ )d γ,xi x j i,j=1 + 1 ( ( ) ( ( ))) ψ dγ + W dγ ψ. ɛ ɛ ɛ Observing tat ( ) n+1 d γ,t d γ γ dγ γ pi p d γ j ( d γ )d γ,xi x j = V γ(ν)h γ =0 i,j=1 onγ(t) and coosing ψ as in (7.4), (7.5) we see tat v is close to being a solution of P (v) = 0 in a neigbourood of 0<t<T Γ(t) {t}.

68 68 K. Deckelnick, G. Dziuk and C. M. Elliott Kinetic anisotropy and a generalized double obstacle pase field model As a generalization of (8.35) we consider te pase field model εβ( ϕ)ϕ t ε DA( ϕ)+ 1 ɛ W (ϕ) =c W ρ(ϕ)g. (8.38) Here, te potential W is taken to be of double obstacle form, were W 0 C 2 [ 1, 1] and W (r) =W 0 (r)+i [ 1,1] (r), (8.39) I [ 1,1] (r) = { + for r > 1, 0 for r 1. A possible example of W 0 is 1 W 0 (r) = 4(1+ξ )[ (r 2 1 ξ 2 ) 2 ξ 4] (8.40) 2 wit ξ (0, ). For ξ =0,W 0 takes te classical smoot double well Ginzburg Landau quadratic form 1 4 (r2 1) 2, wereas for ξ we recover te classical double obstacle potential W (r) = 1 2 (1 r2 )+I [ 1,1] (r). (8.41) Te function ρ is nonnegative and even wit a positive integral across te transition region [ 1, 1]. As above one can sow tat te zero level set of ϕ approximates an interface wic evolves according to te anisotropic forced mean curvature flow: β(ν) γ(ν) V = H γ g. (8.42) Properly (7.9) sould be written as te parabolic variational inequality ɛ β( ϕ)ϕ t (η ϕ)+ɛ DA( ϕ) ( η ϕ)+ 1 W Ω Ω ɛ 0(ϕ)(η ϕ) Ω c W ρ(ϕ)g(η ϕ) for all η K= {H 1 (Ω) : η 1}, (8.43) Ω wic is treated in te viscosity sense in Elliott and Scätzle (1997) because of te singularity in β at te origin. Convergence Te pase field approximation of anisotropic interface motion can be establised as in te isotropic case for smoot potential W (McFadden, Weeler, Braun, Coriell and Sekerka 1993, Weeler and McFadden 1996). Convergence of te double obstacle model to te unique viscosity solution of te anisotropic level set equation was proved in Elliott and Scätzle (1997) even in te case of kinetic anisotropy. Te error bounds for smootly

69 Computation of geometric PDEs and mean curvature flow 69 evolving flows are again O(ɛ 2 ) (Elliott and Scätzle 1996, Elliott, Paolini and Scätzle 1996). Discretization of anisotropic pase field equation Te numerical approximation of (8.43) follows te approac used for te isotropic Allen Can equation (Section 7.3). We use te same notation for te finite element spaces and time discretizations. In order to implement te metod it is necessary to use a regularization β ɛ of β. Fully explicit time-stepping Te fully discrete approximation of (8.43) using explicit time-stepping reads as follows. Algoritm 8. (Anisotropic double obstacle pase field) Let Φ 0 = I ϕ 0.Form =0,...,M 1, find Φ m+1 K suc tat, for all χ K, ɛ(β ε ( Φ m ) Φ m,χ Φ m+1 ) + ɛ(da( Φ m ), χ Φ m+1 ) (8.44) + 1 ε (W 0(Φ m ),χ Φ m+1 ) c W (ρ(φ m )g m,χ Φ m+1 ) 0. Tis sceme is as simple to implement as in te isotropic situation. Let Mβ m, Km, Mρ m,andmw m be defined by (Mβ m ) ij =(β ε ( Φ m )χ i,χ j ), (K m ) ij =(D 2 A( Φ m ) χ i, χ j ), (Mρ m ) j = c W (ρ(φ m )g m,χ j ), (MW m ) j =(W 0(Φ m ),χ j ) for 1 i, j J, 0 m M. Here we made use of te fact tat DA( Φ m )=D 2 A( Φ m ) Φ m, wic follows from (8.4) and te fact tat DA is omogeneous of degree 1. Te variational formulation (8.44) is ten equivalent to te following matrix formulation M m β Φm+1/2 = ( M m β τkm) Φ m + τ ε M m ρ g m τ ε 2 M m W (8.45) and it remains to project Φ m+1/2 component-wise yielding Φ m+1 =PΦ m+1/2. Te use of te mass lumping in (8.45), wic diagonalizes Mβ m, is crucial to eliminate any iteration in solving (8.45). Semi-implicit time-stepping sceme A semi-implicit sceme is obtained by treating te gradient energy term implicitly, yielding te following metod. Algoritm 9. Let Φ 0 = I ϕ 0. For m =0,...,M 1, find Φ m+1 K suc tat, for all χ K, ɛ(β ε ( Φ m ) Φ m,χ Φ m+1 ) + ɛ(da( Φ m+1 ), χ Φ m+1 ) (8.46) + 1 ε (W 0(Φ m ),χ Φ m+1 ) c W (ρ(φ m )g m,χ Φ m+1 ) 0.

70 70 K. Deckelnick, G. Dziuk and C. M. Elliott Te algebraic problem is now a convex optimization problem wit obstacle constraints. Tese scemes are stable in te sense of satisfying energy norm bounds analogous to tose enjoyed by te solution of te PDE. Te stability constraints are analogous to tose olding in te isotropic case. However, owing to te anisotropy in te discrete elliptic operator tere is a lack of a comparison principle wic as proved a barrier to proving convergence. 9. Fourt order flows 9.1. Surface diffusion In tis paragrap we study various ways to approximate surfaces wic evolve according to surface diffusion, tat is, V = Γ H γ on Γ(t). (9.1) Here, H γ denotes te anisotropic mean curvature of te surface Γ(t)asitwas introduced in (8.15). Tis evolution as interesting geometrical properties: if Γ(t) is a closed surface bounding a domain Ω(t), ten te volume of Ω(t) is preserved and te weigted surface area of Γ(t) decreases. At present, te existence and uniqueness teory for surface diffusion is limited to te isotropic case γ(q) = q,q R n+1. For example, it is known tat for closed curves in te plane or closed surfaces in R 3, balls are asymptotically stable subject to small perturbations: see Elliott and Garcke (1997) and Escer, Mayer and Simonett (1998). However, topological canges suc as pinc-off are possible (Giga and Ito 1998, Mayer and Simonett 2000) and a one-dimensional grap may lose its grap property. An example of pinc-off is sown in Figure 9.1. We start wit te axially symmetric initial surface given by r 0 (x) =1+0.05(sin (5.5x)+sin(5x)), x (0, 8π). (9.2) Pinc-off appens after a very long computation time. Note te different scaling of te x- and te r-axis. Tis example was first computed in Coleman, Falk and Moaker (1995) Surface diffusion for axially symmetric surfaces In applications one is interested in te stability of so-called wiskers, wic are axially symmetric cylindrical bodies of small diameter wit respect to teir lengt: see Nicols and Mullins (1965) and Coleman, Falk and Moaker (1995). Let us consider an axially symmetric cylindrical body, wose boundary Γ(t) ={x R 3 x =(x, r(x, t)cosφ, r(x, t)sinφ),x [0,L],φ [0, 2π]}

71 Computation of geometric PDEs and mean curvature flow 71 r t =0.0 t =10.0 t =20.0 t =28.0 t = x Figure 9.1. Evolution of te initial surface given by (9.2) for t =0.0, 10.0, 20.0, 28.0 and28.2. evolves by V = Γ H. We assume tat te radius r is a smoot positive function, wic is periodic in x, so tat r(0,t) = r(l, t). In tese coordinates te mean curvature of Γ(t) is 1 H = r 1+rx 2 r xx 1+r 2 3 = 1 x r 1+rx 2 ( ) rx 1+r 2 x x, (9.3) wile normal velocity and te surface Laplacian, respectively, are given by ( ) r t 1 V =, Γ H = 1+r 2 x r rhx. 1+rx 2 1+r 2 x x

72 72 K. Deckelnick, G. Dziuk and C. M. Elliott It follows from tese two equations tat r satisfies te quasilinear fourt order parabolic problem r t = 1 ( ) rhx r 1+r 2 x x in I (0,T], (9.4) r(0,t)=r(l, t) in (0,T], (9.5) H(0,t)=H(L, t) in (0,T], (9.6) r(, 0) = r 0 in I, (9.7) were I =(0,L)andH is given by (9.3). Te initial function r 0 is assumed to be periodic and positive. For discretization purposes it is convenient to split te fourt order problem into two coupled second order equations for te radial variable r and te mean curvature H resulting in te following variational form: I rhζ dx I I rr t η dx + 1+r 2 x ζ dx I I rh x η x 1+r 2 x dx =0 (9.8) rr x ζ x 1+r 2 x dx =0 (9.9) for all η, ζ Hper(I) 1 ={η H 1 (I) η(0) = η(l)}. We note tat Coleman, Falk and Moaker (1995) proposed a similar second order splitting and used R = r 2 and H as te variables. We employ (9.8), (9.9) in order to define a semidiscrete sceme using linear finite elements to approximate r and H. Let 0 = x 0 <x 1 < < x N = L, j = x j x j 1 and = max 1 j N j. We sall make an inverse assumption of te form ρ j for all j = 1,...,N, were ρ > 0is independent of. Te spatial discretization is based on piecewise linear finite elements, X = {φ C 0 (Ī) φ [x j 1,x j ] P 1, 1 j N,φ (0) = φ (L)}. Our discrete problem now reads: find r,h :[0,T] X suc tat r H,x η,x r r,t η dx + dx =0, (9.10) I I 1+r,x 2 I r H ζ dx 1+r,x 2 ζ dx I I r r,x ζ,x dx = 0 (9.11) 1+r,x 2 for all η,ζ X, t [0,T] and wit r (0) = I r 0, were I denotes te Lagrange interpolation operator. In Deckelnick, Dziuk and Elliott (2003a) a convergence analysis for te above sceme is carried out. Te principal results are error bounds for position and mean curvature as described in te following teorem.

73 Computation of geometric PDEs and mean curvature flow 73 Teorem 9.1. Let us assume tat (9.4) (9.7) as a sufficiently smoot positive solution on a maximal time interval [0,T max ). Ten te discrete solution (r,h ) exists on [0,T] for all T < T max and tere is an 0 > 0 suc tat, for all 0 < 0, T sup r r 2 H 1 (I) + H H 2 H 1 (I) dt C2. (9.12) (0,T ) Surface diffusion for graps Te anisotropic surface diffusion (1.8) of a grap Γ(t) ={(x, u(x, t)) x Ω} sitting above a domain Ω R n leads to a igly nonlinear fourt order geometric partial differential equation. For graps te Laplace Beltrami operator applied to anisotropic mean curvature H γ reads Γ H γ = 1 ( ) Q (Q u u I ) H Q 2 γ, (9.13) were we ave again written Q = 1+ u 2. Recalling (8.17) as well as V = ut Q, we see tat (1.8) for graps is equivalent to te partial differential equation ( ( u t + Q I u Q u ) ( n )) γ pi p Q j ( u, 1)u xi x j =0. (9.14) i,j=1 As in te previous section, it is convenient to split te fourt order problem into two second order problems as follows: ( ( u t = Q I u Q u ) ) w, (9.15) Q n w = γ pi p j ( u, 1)u xi x j. (9.16) i,j=1 Te system is closed using Neumann boundary conditions and an initial condition for u: ( Q I u Q u ) w ν Ω =0, (9.17) Q D γ ( u, 1) (ν Ω, 0) = 0, (9.18) u(, 0) = u 0. (9.19) Te first equation, (9.17), is te zero mass flux condition, wereas te second equation, (9.18), is te natural variational boundary condition wic defines w as te variational derivative or cemical potential for te surface energy functional. Note tat an initial condition on w is not required. Te

74 74 K. Deckelnick, G. Dziuk and C. M. Elliott problem (9.15) (9.18) can easily be rewritten in variational form, namely ( u t η + Q I u Ω Ω Q u ) w η =0, (9.20) Q n wψ γ pj ( u, 1)ψ xj = 0 (9.21) Ω j=1 for all η, ψ H 1 (Ω),t (0,T]. Replacing H 1 (Ω) by te space X of piecewise linear finite elements we immediately arrive at a natural way to discretize in space. A finite element error analysis for te resulting semidiscrete sceme in te isotropic case was carried out by Bänsc, Morin and Nocetto (2004) for graps in arbitrary space dimension. Te time discretization follows te ideas of te discretization tecniques introduced in Algoritm 7 and Teorem 8.7, leading to te following. Ω Algoritm 10. (Anisotropic surface diffusion of graps) Let τ>0 be te time-step size wit Mτ = T and assume tat λ>0isasinteorem 8.7. Let te initial value u 0 X. For m =1,...,M, compute u m+1,w m+1 1 τ (u m+1 Ω Ω X suc tat u m )η + w m+1 ψ λ τ Ω Q m Ω Q m Ω ( I um Q m um Q m γ( u m, 1) (Q m (u m+1 )2 u m ) ψ i=1 ( I um Q m Ω γ Pi ( u m, 1)ψ x i um Q m ) w m+1 η =0, ) (u m+1 u m ) ψ =0 for all discrete test functions η,ψ X. Here Q m = 1+ u m 2. Note tat in eac time-step a linear system of equations as to be solved. An error analysis for tis sceme is carried out in Deckelnick, Dziuk and Elliott (2003b). Teorem 9.2. Let u be a sufficiently smoot solution of anisotropic surface diffusion (9.14), (9.17) (9.19) on te domain Ω (0,T)andsetw = H γ. Let X be te space of continuous piecewise linear finite elements.

75 Computation of geometric PDEs and mean curvature flow 75 Figure 9.2. Anisotropic surface diffusion wit a very strong anisotropy. Level lines are sown for te time-steps 0, 10, 200 and a view from a position vertically above te grap for time-step 300. Ten for te discrete solution u m,wm estimates M max m=1,...,m um u m 2 L 2 (Ω) + τ (u m u m max ) 2 m=1,...,m Ω Q m + τ m=1 M m=1 (m =1,...,M) we ave te error Ω w m w m 2 L 2 (Ω) c(τ ), (w m w m ) 2 Q m 1 c(τ ).

76 76 K. Deckelnick, G. Dziuk and C. M. Elliott Here u m = u(,mτ), w m = w(,mτ). Te proof uses ideas tat were developed for te motion of graps by anisotropic mean curvature. Tere is neiter a restriction on te space dimension nor a coupling of time-step size and grid size. In two dimensions inverse estimates yield L ((0,T),H 1 (Ω))-convergence for u and convergence in L 2 ((0,T),H 1 (Ω)) for w. In Figure 9.2 we sow computational results for anisotropic surface diffusion of a grap. Te anisotropy is cosen to be a regularized l 1 norm (see Figure 8.1), 3 γ(p) = p 2 j + ε2 p 2 (9.22) j=1 wit ε =10 3. Tus te Frank diagram is a smooted octaedron and te Wulff sape is a smooted cube. Te initial data were taken to depend on tree random numbers r 1,r 2,r 3 (0, 1), u 0 (x) =0.25 (sin (2πr 1 x 1 )+0.25 sin (3πr 2 x 2 )) (0.1sin(2πr 3 x 1 )+sin(5πr 1 x 2 )) sin (2πr 2 x 1 x 2 ). (9.23) We used Neumann boundary conditions and te rigt-and side (for te curvature equation) f =1 x 2 1 x2 2. Te level lines for some time-steps are sown in Figure 9.2. Te Wulff sape (a smoot cube) appears in te solution as a consequence of te rigt-and side f. For more computational results we refer to Deckelnick, Dziuk and Elliott (2003b) Pase field model for surface diffusion Just as te pase field model for mean curvature flow is gradient flow for te gradient energy functional and leads to a second order parabolic equation, a pase field model for surface diffusion may also be based on te same energy functional and a suitable approximation of te Laplace Beltrami operator leading to a nonlinear degenerate fourt order parabolic equation. Te appropriate setting is in te context of te Can Hilliard equation for pase separation in binary alloys. Te pase function ϕ may be viewed as te difference in mass fractions of te two species so tat te values ϕ = ±1 are associated wit te pure materials. Stable pases of te alloy are ten associated wit te minima of a double well bulk energy W, wic in te regular solution form is W (ϕ) = θ 2 [(1 + ϕ)ln[1+ϕ]+(1 ϕ)ln[1 ϕ]] (1 ϕ2 ). Tis omogeneous free energy function is non-convex wit a double well, for θ < 1, and W (ϕ) is said to be te omogeneous cemical potential.

77 Computation of geometric PDEs and mean curvature flow 77 Te Can Hilliard gradient energy functional is ten [ ε E(ϕ) = Ω 2 ϕ ] ε W (ϕ). (9.24) Te functional derivative of tis energy is used to define te cemical potential w = ɛ ϕ + W (ϕ). (9.25) ɛ Mass conservation is t ϕ + J =0, (9.26) were J is te mass flux and typically for diffusion J = M(ϕ) w (9.27) wit te degenerate mobility {M}(ϕ) =1 ϕ 2. Te upsot is a fourt order Can Hilliard equation wit degenerate mobility. Interface asymptotics (Can, Elliott and Novick-Coen 1996) sow tat, as θ(ɛ) andɛ tend to zero, te zero level set of ϕ approximates a surface evolving by surface diffusion. Computational results in a setting wic includes a forcing due to an electric field may be found in Barrett, Nürnberg and Styles (2004) Willmore flow Our starting point is te Willmore functional E(X) = 1 H 2 da, Γ=X(M), (9.28) 2 Γ were M is an n-dimensional reference manifold and X : M R n+1 is a smoot immersion. Considering variations X ɛ (p) =X(p) +ɛφ(p),p M, were φ : M R n+1 is smoot and vanises near M, one obtains te formula E (X),φ = d dɛ E(X ɛ) ɛ=0 (9.29) = Γ X ( Γ φ +2ν Γ ν Γ φ ) + 1 H 2 Γ X Γ φ Γ 2 Γ = Γ (Hν) Γ φ 2 H Γ ν Γ φ + 1 H 2 Γ X Γ φ, Γ Γ 2 Γ were we ave used (2.10). Note tat Γ X Γ φ = +1 j,k=1 D jx k D j φ k. Willmore flow ten arises as te L 2 -gradient flow of te Willmore functional, tat is, X t φ da = E (X),φ. (9.30) Γ

78 78 K. Deckelnick, G. Dziuk and C. M. Elliott Figure 9.3. Surface relaxing under Willmore flow, cut open at x 2 =0. Using integration by parts one obtains te nonlinear evolution equation of fourt order, tat is, X t = Γ (Hν) 2 Γ (H Γ ν ) + H Γ H 1 2 H3 ν. (9.31) If we take te scalar product of te above expression wit ν and observe tat Γ ν ν = Γ ν 2, we obtain te evolution law V = Γ H + H Γ ν H3 on Γ(t). (9.32) Note tat from Section 2.3 we ave Γ ν 2 = n+1 j,k=1 (D j ν k ) 2 = κ κ 2 n. For two-dimensional surfaces Γ we ten ave Γ ν 2 = κ κ 2 2 =(κ 1 + κ 2 ) 2 2κ 1 κ 2 = H 2 2K wit Gauss curvature K. Tis leads to te evolution law V = Γ H H3 2KH. (9.33) Compared wit te surface diffusion problem (1.7) additional dimensiondependent nonlinearities appear. Up to now analytical results for te above evolution law ave been primarily obtained for te case of closed surfaces. In Simonett (2001) it is sown tat a unique local solution of (9.32) satisfying Γ(0) = Γ 0 exists provided tat Γ 0 is a compact closed immersed and orientable C 2,α -surface in R 3. Te solution exists globally in time if Γ 0 is sufficiently close to a spere in C 2,α. Using different metods, Kuwert and Scätzle (2004a) obtain global existence of solutions provided tat Γ 0 A 2 is sufficiently small, were A denotes te trace-free part of te second fundamental form. Tey were subsequently able to remove te smallness

79 Computation of geometric PDEs and mean curvature flow 79 assumption and to prove te existence of a global smoot solution provided tat E(X 0 ) 16π, were Γ 0 = X 0 (S 2 ) (see Kuwert and Scätzle (2004b) and note tat our definition differs from teirs by a factor of 2). Tere is numerical evidence (Mayer and Simonett 2002) tat te above condition is optimal in te sense tat te flow develops a singularity if te initial surface as energy greater tat 16π. A major problem in te numerical solution of tis problem is te treatment of Gauss curvature, wic is a nonlinear expression of te principal curvatures and contrary to mean curvature is not easily accessible to variational metods. Te elastic flow of curves Let us start wit te one-dimensional parametric problem. Te Bernoulli model of an elastic rod (Truesdell 1983) described by a closed curve X : S 1 R 2 uses te curvature integral (9.28) as elastic energy. Since tis energy can be minimized by scaling, one usually adds lengt multiplied by a parameter λ>0 resulting in te functional E λ (X) = 1 H 2 ds + λ 1ds. 2 Γ Let us introduce Y = Hν, were H is just te usual curvature of a curve. We ten obtain from (9.29) and te Frenet formula Γ ν = Hτ τ (wit te unit tangent τ) E λ (X),φ = Γ Γ Y Γ φ ds 2 H Γ ν Γ φ Γ H 2 Γ X Γ φ + λ Y φ Γ Γ H 2 Γ X Γ φ + λ = Γ Y Γ φ 3 Y φ. Γ 2 Γ Γ Tus one may expect te gradient flow for E λ to be given by te equation X t = Γ Y 3 2 Γ (H 2 Γ X ) λy. (9.34) Long time existence for tis problem as been proved by Polden (1996). Just as in Section 4.1 we tink of X as a mapping from R [0,T)intoR 2. We ten ave te following system to be satisfied by X and Y : X t 1 X θ ( Yθ X θ ) θ 2 X θ ( Y 2 X θ X θ Y + 1 X θ Γ ) θ + λy =0, (9.35) ) = 0 (9.36) ( Xθ X θ in [0, 2π] (0,T). In addition, X as to satisfy te initial condition X(, 0) = X 0 in I =[0, 2π] and te periodicity condition X(θ, t) =X(θ +2π, t) for θ

80 80 K. Deckelnick, G. Dziuk and C. M. Elliott Figure 9.4. Time series of te two-dimensional lengt-preserving elastic flow (grapically scaled).

81 Computation of geometric PDEs and mean curvature flow 81 0 t<t, θ R. As in te derivation of (4.6) we obtain a variational formulation of (9.35), (9.36), Y θ ϕ θ X t ϕ X θ + 3 Y 2 X θ ϕ θ + λ Y ϕ X θ =0, I I X θ 2 I X θ I X θ ψ θ Y ψ X θ =0 I I X θ for all test functions ϕ, ψ Hper([0, 1 2π]; R 2 ). We use tis weak form of our problem for a finite element discretization in space, wic in one space dimension leads to a suitable difference sceme. Te derivation of tis sceme follows te derivation of (4.9), additionally using mass lumping in bot equations. Let us denote by φ 1,...,φ N te basis of te finite element space S introduced in Section 4.1. Ten, expanding X (θ, t) = N j=1 X j(t)φ j (θ), Y (θ, t) = N j=1 Y j(t)φ j (θ) wit vectors X j (t),y j (t) R 2, yields te following system of 2N ordinary differential equations: 1 2 (q j + q j+1 ) ( ) Y j Y j 1 Ẋ j + λy j + Y j+1 Y j (9.37) q j q j+1 p j X j X j 1 q j + p j+1 X j+1 X j q j+1 =0 1 2 (q j + q j+1 )Y j X j X j 1 + X j+1 X j = 0 (9.38) q j q j+1 (j =1,...,N), were X 0 = X N,X N+1 = X 1,Y 0 = Y N,Y N+1 = Y 1, and te initial values are given by X j (0) = X 0 (θ j )(j =1,...,N). Furtermore, q j = X j X j 1, p j = 1 2( Yj Y j 1 Y j + Y j 2). (9.39) A more detailed description can be found in Dziuk, Kuwert and Scätzle (2002). Te paper actually treats curves in arbitrary codimension bot sowing long time existence of solutions as well as numerical examples. We include ere a computation wic sows te unravelling of a planar knotted curve under te lengt-preserving elastic flow in Figure 9.4. Parametric Willmore flow of surfaces Te equation for Willmore flow of two-dimensional surfaces in R 3 is muc more difficult to treat. Tis is because Gauss curvature appears in te equation (9.33) for Willmore flow. Mean curvature H is given as a divergence expression (see (2.9)), so tat in te discretization of parametric mean curvature flow, for example, we were able to formulate te mean curvature vector in a weak form, wic ten lead to a finite element sceme for parametric mean curvature flow. We were able to define te mean curvature vector of a polyedron as a continuous and piecewise linear vector-valued

82 82 K. Deckelnick, G. Dziuk and C. M. Elliott Figure 9.5. Half of a spere eversion: te unravelling of a perturbed Willmore spere under parametric Willmore flow (scaled grapically). Time-steps 0, 5600, 6000, 6400, 6800, 7000, 7200, 7400, 7600 and 8000.

83 Computation of geometric PDEs and mean curvature flow 83 function. Rusu (2001) employed a trick to remove Gauss curvature from te equations. Let us briefly describe te underlying idea, wic we tink is very important for applications of Willmore flow. For simplicity we look at closed surfaces. Going back to (9.29) and introducing te mean curvature vector Y = Hν as a new variable, we ave E (X),φ = Γ Y Γ φ 2 Y ν Γ ν Γ φ + 1 Y 2 Γ X Γ φ. Γ Γ 2 Γ Integration by parts gives ( Y ν Γ ν Γ φ = Γ Y Γ φ ( ν Γ Y ) (ν Γ φ )). Γ Γ If we insert tis identity into te above expression for E (X) we obtain E (X),φ = R(ν) Γ Y Γ φ + 1 Y 2 Γ X Γ φ Γ 2 Γ wit te reflection matrix R kl (ν) =δ kl 2ν k ν l. Starting from (9.30) we can now write down a variational formulation for parametric Willmore flow wic uses position X and mean curvature vector Y as variables: find X : M [0,T) R 3 suc tat X t φ da R(ν) Γ Y Γ φ da + 1 Y 2 Γ X Γ φ da =0, Γ Γ 2 Γ Y ψ da Γ X Γ ψ da =0 Γ for all test functions φ, ψ H 1 (Γ) 3. Here, Γ = Γ(t) =X(M,t). Furtermore we require te initial condition X(, 0) = X 0. We observe tat all quantities are well defined for a polyedral surface Γ so tat it is possible to use tis formulation in order to approximate solutions by linear finite elements (see Rusu (2001) for more details and Clarenz, Diewald, Dziuk, Rumpf and Rusu (2004) for applications to problems in image restoration) Willmore flow of graps If te two-dimensional surface Γ = {(x, u(x, t)) x Ω} is a grap above some domain Ω R 2, ten we can directly derive a fourt order parabolic partial differential equation for u. We write te equation (9.33) for a grap. In order to write down tis equation we note tat te quantities V, H, K and Γ H appearing in (9.33) are expressed in terms of u as in (5.1) and K = detd2 u Q 4, (9.40) Γ H = 1 ( Q (Q I u Q u ) ) H. (9.41) Q Γ

84 84 K. Deckelnick, G. Dziuk and C. M. Elliott We can rewrite te last equation as ( ( ) 1 u u Γ H = I Q Q 2 (QH) Wit te expression (5.1) for H we conclude ( I 1 Q u u Q 2 ) Q = 1 Q ) ( ( 1 H I Q ( Q u Q u ) ) ) u u Q 2 Q. (9.42) + H u Q, (9.43) and a calculation sows tat ( ( 1 Q u )) Q Q u = 2K. (9.44) Inserting (9.43) and (9.44) into (9.42), we obtain ( ( ) ) ( 1 u u Γ H = I Q Q 2 (QH) +2HK H H u ) Q ( ( ) ) 1 u u = I Q Q 2 (QH) +2HK 12 ( ) H 2 Q u 1 2 H3. Comparing tis expression wit (9.33), we obtain a fourt order parabolic partial differential equation for u, ( ( ) 1 u u u t +Q I Q Q 2 (QH) ) 12 ( ) H 2 Q Q u =0 inω (0,T). (9.45) As before we can split te fourt order problem into two second order equations. Te above equation suggests using te eigt u and w = QH as variables wic is different from te case of surface diffusion. Note tat Gauss curvature no longer appears. Te above ideas were introduced by Droske and Rumpf (2004) for a level set approac to Willmore flow. Te finite element approac is now based on dividing (9.45) by Q, multiplying by a test function ϕ H0 1 (Ω) and integrating by parts. Tis leads to ( ) u t ϕ Ω Q + 1 u u I Ω Q Q 2 w ϕ + 1 w 2 u ϕ =0, (9.46) 2 Ω Q3 wζ Ω Q u ζ =0, (9.47) Ω Q for all ϕ, ζ H0 1 (Ω). As boundary conditions we coose u = u 0 on Ω [0,T] Ω {0}, (9.48) w =0 on Ω [0,T] (9.49)

85 Computation of geometric PDEs and mean curvature flow 85 wit a given function u 0, wic is independent of time. For te error estimates we need te following regularity of te continuous solution: k u t k L ((0,T); H 4 2k, (Ω)) L 2 ((0,T); H 5 2k (Ω)), k =0, 1, 2. (9.50) Tus we need ig compatibility of initial and boundary data. Te spatially discrete problem now reads as follows. Find (u (t),w (t)), 0 t T,suc tat u (t) I u 0 X 0, w (t) X 0, u (0) = u 0 X 0 and ( u t ϕ 1 + I u ) u Ω Q Ω Q Q 2 w ϕ + 1 w 2 u ϕ =0 forallϕ x 0, 2 Ω Q 3 w ζ u ζ =0 forallζ x 0. Ω Q Ω Q Te discrete initial value u (, 0) = u 0 X is cosen as te minimal surface projection Ω u 0 ζ 1+ u0 = 2 of te continuous initial value u 0. Ω u 0 ζ 1+ u0 2 for all ζ X 0, (9.51) Teorem 9.3. Let us assume tat (9.45), (9.48) as a unique solution u on te interval [0,T], wic satisfies (9.50). Also suppose tat u 0 is defined as te projection (9.51) of u 0.Ten sup (u u )(t) + sup (w w )(t) c 2 log 2, (9.52) 0 t T 0 t T sup (u u )(t) c, (9.53) Appendix 0 t T T T 0 0 u t u t 2 dt c 4 log 4, (9.54) (w w ) 2 dt c 2. (9.55) Proof of Lemma 2.1. We prove (2.23), leaving (2.22) to te reader. Fix t 0 (0,T). For x Γ(t 0 ) let U x,δ x > 0andu be as in (2.19). By te implicit function teorem tere exists an open set Ũx U x,0< δ x δ x suc tat Ũx (t 0 δ x,t 0 + δ x ) Q and Ũx Γ(t) can be written as a grap over some open set Ω x R n for t t 0 < δ x. Since Γ(t 0 ) x Γ(t0 )Ũx and

86 86 K. Deckelnick, G. Dziuk and C. M. Elliott Γ(t 0 ) is compact, tere exist a 1,...,a N wit Γ(t 0 ) N j=1ũj, Ũj = Ũa j.let Q j = Ũj (t 0 δ j,t 0 + δ j ) and let η j C0 (Q j), 1 j N, be a partition of unity wic satisfies N j=1 η j(x, t) =1for(x, t) in a neigbourood of Γ(t 0 ) {t 0 }.Fort close to t 0 we ten ave d g(x, t)da = dt Γ(t) N j=1 d η j (x, t)g(x, t)da. (9.56) dt Ũ j Γ(t) Let us fix j {1,...,N}; by construction tere exists Ω R n and v C 2,1 (Ω (t 0 δ j,t 0 + δ j )) suc tat w.l.o.g. Ũ j Γ(t) ={(x,v(x,t)) x =(x 1,...,x n ) Ω}. Abbreviating = η j g,weave (x, t)da = Ũ j Γ(t) Ω (x,v(x,t),t) 1+ x v 2 dx so tat we obtain, for t close to t 0, d ( ) da = xn+1 v t + t 1+ x v dt 2 + x v x v t Ũ j Γ(t) Ω Ω 1+ x v 2 ( ( ) = xn+1 v t + t 1+ x v 2 x v x )v t Ω Ω 1+ x v 2 ( x + xn+1 x v ) x v 1+ x v v 2 t Ω were we ave used integration by parts observing ( tat) supp (,t) Ω. Recalling tat ν = ( x v, 1) and H = 1+ x v 2 x x v we finally get 1+ x v 2 ( ) d da = dt Ũ j Γ(t) Ũ j Γ(t) ν V + t + V H da. Note tat te above identity as been derived under te implicit assumption tat x v t exists; te general case can be justified wit te elp of an approximating argument. If we return to (9.56) and recall tat N j=1 η j 1 in a neigbourood of Γ(t 0 ) {t 0 }, we obtain (2.23) at t = t 0. Acknowledgements We would like to tank Micael Fried and Alfred Scmidt for providing figures of calculations for te Stefan problem wit undercooling and Vanessa Styles for providing figures of pase field calculations. Te work was supported by te Deutsce Forscungsgemeinscaft via DFG-Forscergruppe Nonlinear partial differential equations: Teoretical and numerical analysis and

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