A New Cement to Glue Nonconforming Grids with Robin Interface Conditions: The Finite Element Case


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1 A New Cement to Glue Nonconforming Grids wit Robin Interface Conditions: Te Finite Element Case Martin J. Gander, Caroline Japet 2, Yvon Maday 3, and Frédéric Nataf 4 McGill University, Dept. of Matematics and Statistics, Montreal (ttp:// 2 Université Paris 3, Laboratoire d Analyse, Géométrie et Applications 3 Université Pierre et Marie Curie, Laboratoire Jacques Louis Lions (ttp:// 4 CNRS, UMR 764, CMAP, Ecole Polytecnique, 928 Palaiseau (ttp:// Summary. We present and analyze a new nonconforming domain decomposition metod based on a Scwarz metod wit Robin transmission conditions. We prove tat te metod is well posed and convergent. Our error analysis is valid in two dimensions for piecewise polynomials of low and ig order and also in tree dimensions for P elements. We furter present an efficient algoritm in two dimensions to perform te required projections between arbitrary grids. We finally illustrate te new metod wit numerical results. Introduction We propose a domain decomposition metod based on te Scwarz algoritm tat permits te use of optimized interface conditions on nonconforming grids. Suc interface conditions ave been sown to be a ey ingredient for efficient domain decomposition metods in te case of conforming approximations (see Després [99], Nataf et al. [995], Japet [998], Cevalier and Nataf [998]). Our goal is to use tese interface conditions on nonconforming grids, because tis simplifies greatly te parallel generation and adaptation of meses per subdomain. Te mortar metod, first introduced in Bernardi et al. [994], also permits te use of nonconforming grids, and it is well suited to te use of DiricletNeumann (Gastaldi et al. [996]) or NeumannNeumann metods applied to te Scur complement matrix. But te mortar metod can not be used easily wit optimized transmission conditions in te framewor of Scwarz metods. In Acdou et al. [22], te case of finite volume discretizations as been introduced and analyzed. Tis paper is a first step in te finite element case; we consider only interface conditions of order ere.
2 26 Martin J. Gander, Caroline Japet, Yvon Maday, and Frédéric Nataf 2 Definition of te metod and te iterative solver We consider te model problem (Id )u = f in Ω, u = on Ω, () were f is given in L 2 (Ω) and Ω is a C, (or convex) domain in IR d, d = 2 or 3. We assume tat it is decomposed into K nonoverlapping subdomains Ω = K Ω, were Ω, K are C, or convex polygons in two or polyedrons in tree dimensions. We also assume tat tis domain decomposition is conforming. Let n be te unit outward normal for Ω and Γ,l = Ω Ω l. Te variational statement of problem () consists of writing te problem as follows: Find u H (Ω) suc tat ( u v + uv)dx = fvdx, v H (Ω). (2) Ω Ω We introduce now te space H (Ω ) = {ϕ H (Ω ), ϕ = over Ω Ω }, and te constrained space K K V ={(v,q) ( H (Ω )) ( H /2 ( Ω )), v =v l and q = q l on Γ,l }. Problem (2) is ten equivalent to te following: Find (u,p) V suc tat Ω ( u v + u v )dx = H /2 ( Ω ) < p, v > H /2 ( Ω ) Ω f v dx, v K H (Ω ). Being equivalent wit te original problem, were p = u n over Ω, tis problem is naturally well posed. We now describe te iterative procedure in te continuous case, and ten its discrete, nonconforming analog. 2. Te continuous case We introduce for α IR, α >, te zerot order transmission condition p + αu = p l + αu l over Γ,l and te following algoritm: let (u n, pn ) H (Ω ) H /2 ( Ω ) be an approximation of (u, p) in Ω at step n. Ten, (u n+, p n+ ) is te solution in H (Ω ) H /2 ( Ω ) of
3 Nonconforming Grids wit Robin Interface Conditions 26 ( u n+ v + u n+ ) v dx H /2 ( Ω ) < p n+, v > H /2 ( Ω ) Ω = f v dx, v H (Ω ), Ω < p n+ +αu n+, v > Γ,l=< p n l +αun l, v > Γ,l, v H /2 (Γ,l ). Convergence of tis algoritm is sown in Després [99] using energy estimates and summarized in te following Teorem. Assume tat f is in L 2 (Ω) and (p ) K l H/2 (Γ,l ). Ten, algoritm (3) converges in te sense tat lim u n n ( u H (Ω ) + p n p ) H /2 ( Ω ) =, for K, were u solves (), u = u Ω, p = u n on Ω for K. 2.2 Te discrete case We introduce now te discrete spaces: eac Ω is provided wit its own mes T, K, suc tat Ω = T T T. For T T, let T be te diameter of T and te discretization parameter, = max K (max T T T ). Let ρ T be te diameter of te circle in two dimensions or spere in tree dimensions inscribed in T. We suppose tat T is uniformly regular: tere exists σ and τ independent of suc tat T T, σ T σ and τ T. We consider tat te sets belonging to te meses are of simplicial type (triangles or tetraedra), but te following analysis can be applied as well for quadrangular or exaedral meses. Let P M (T) denote te space of all polynomials defined over T of total degree less tan or equal to M for our Lagrangian finite elements. Ten, we define over eac subdomain two conforming spaces Y and X by Y = {v, C (Ω ), v, T P M (T), T T }, X = {v, Y, v, Ω Ω = }. (4) Te space of traces over eac Γ,l of elements of Y is denoted by Y,l. In te sequel we assume for te sae of simplicity tat referring to a pair (, l) implies tat Γ,l,l is not empty. Wit eac suc interface we associate a subspace W of Y,l lie in te mortar element metod; for two dimensions, see Bernardi et al. [994], and for tree dimensions see Belgacem and Maday [997] and Braess and Damen [998]. To be more specific, we recall te situation in two dimensions: if te space X consists of continuous piecewise polynomials of degree M, ten it is readily noticed tat te restriction of X,l to Γ consists of finite element functions adapted to te (possibly curved) side Γ,l of piecewise polynomials of degree M. Tis side as two end points wic we denote by x,l and x,l n and wic belong to te set of vertices of te (3)
4 262 Martin J. Gander, Caroline Japet, Yvon Maday, and Frédéric Nataf corresponding triangulation of Γ,l : x,l, x,l,...,x,l is ten te subspace of tose elements of Y,l M over bot [x,l, x,l n, x,l,l n. Te space W tat are polynomials of degree ] and [x,l n, x,l n ]. As before, te space W,l product space of te W over eac l suc tat Γ,l. Te discrete constrained space is ten defined by K K V = {(u,p ) ( X) ( W ), is te,l ((p, + αu, ) ( p,l + αu,l ))ψ,,l =, ψ,,l W }, Γ,l and te discrete problem is te following: Find (u,p ) V suc tat v = (v,,...v,k ) K X, ( u, v, +u, v, ) dx p, v, ds= f v, dx. (5) Ω Ω Ω Te discrete algoritm is ten as follows: let (u n,, pn, ) X W be a discrete approximation of (u,p) in Ω at step n. Ten, (u n+ ) is te,, pn+, solution in X W of ( ) u n+, v,+u n+, v, dx p n+, v,ds= f v, dx, v, X, (6) Ω Ω Ω (p n+, + αun+, )ψ,,l = ( p n,l + αu n,l,l)ψ,,l, ψ,,l W. (7) Γ,l Γ,l Remar. Let π,l denote te ortogonal projection operator from L 2 (Γ,l ) onto. Ten (7) corresponds to W,l p n+, + απ,l(u n+, ) = π,l( p n,l + αun,l ) over Γ,l. (8) Remar 2. A fundamental difference between tis metod and te original mortar metod in Bernardi et al. [994] is tat te interface conditions are cosen in a symmetric way: tere is no master and no slave, see also Gander et al. [2]. Equation (8) is te transmission condition on Γ,l for Ω, and te transmission condition on Γ,l for Ω l is p n+,l + απ l, (u n+,l ) = π l,( p n, + αun, ) over Γ,l. (9) In order to analyze te convergence of tis iterative sceme, we define for any p in K L2 ( Ω ) te norm p 2, = ( K l= l p 2 ) H 2, 2 (Γ,l )
5 were. H 2 (Γ,l ) Nonconforming Grids wit Robin Interface Conditions 263 stands for te dual norm of H 2 (Γ,l ). Convergence of te algoritm (6)(7) can be sown again using an energy estimate, see Japet et al. [23]. Teorem 2. Assume tat α c for some constant c small enoug. Ten, te discrete problem (5) as a unique solution (u,p ) V. Te algoritm (6)(7) is well posed and converges in te sense tat lim n ( un, u, H (Ω ) + p n, p, ) =, for K. H 2 (Γ,l ) l 3 Best approximation properties In tis part we give best approximation results of (u,p) by elements in V. Te proofs can be found in Japet et al. [23] for te two dimensional case wit te degree of te finite element approximations M 3 and in tree dimensions for first order approximations. Teorem 3. Assume tat te solution u of () is in H 2 (Ω) H (Ω) and u = u Ω H 2+m (Ω ) wit M m, and let p,l = u n over eac Γ,l. Ten, tere exists a constant c independent of and α suc tat u u + p p 2, c(α2+m + +m ) + c( m α + +m ) u H 2+m (Ω ) p,l. H +m 2 (Γ,l ) Assuming more regularity on te normal derivatives on te interfaces, we ave Teorem 4. Assume tat te solution u of () is in H 2 (Ω) H (Ω) and u = u Ω H 2+m (Ω ) wit M m, and p,l = u n is in H 3 2 +m (Γ,l ). Ten tere exists a constant c independent of and α suc tat u u + p p 2, c(α 2+m + +m ) + c( +m α + 2+m )(log ) β(m) l u H 2+m (Ω ) p,l 3. H +m 2 (Γ,l ) Remar 3. Te Robin parameter α can depend on in te previous teorems, lie te optimal Robin parameter α opt in section 5. l
6 264 Martin J. Gander, Caroline Japet, Yvon Maday, and Frédéric Nataf 4 Efficient projection algoritm Te projection (8) between non conforming grids is not an easy tas in an algoritm, already for two dimensional problems, since one needs to find te intersections of corresponding arbitrary grid cells. A sort and efficient algoritm as been proposed in Gander et al. [2] in te finite volume case wit projections on piecewise constant functions. In our case, we denote by n te dimension of W,l, and we introduce te sape functions {ψ,l i } i n of W,l. Ten, to compute te rigt and side in (7), we need to compute te interface matrix M = ( ψ,l i Γ,l ψ l, j ) i,j n. In te same spirit as in Gander et al. [2], te following sort algoritm in Matlab computes te interface matrix M for nonmatcing grids in one pass. function M=InterfaceMatrix(ta,tb); n=lengt(tb); m=lengt(ta); ta(m)=tb(n); % must be numerically equal j=; M=zeros(n,lengt(ta)); for i=:n, tm=tb(i); wile ta(j+)<tb(i+), M(i:i+,j:j+)=M(i:i+,j:j+)+intMortar(ta(j),ta(j+),... tb(i),tb(i+),tm,ta(j+),j== j==m,i== i==n); j=j+; tm=ta(j); end; M(i:i+,j:j+)=M(i:i+,j:j+)+intMortar(ta(j),ta(j+),... tb(i),tb(i+),tm,tb(i+),j== j==m,i== i==n); end; It taes two vectors ta and tb wit ordered entries, wic represent two nonmatcing grids at te interface, wit ta()=tb(), ta(end)=tb(end), and computes te matrix M(i,j)= Γ,l b i a j, were b i is te at function for te node tb(i) and a j is te at function for te node ta(j). Te mortar condition of constant sape functions at te corners is taen into account, and from te resulting matrix M te first and last row and column needs to be removed. Tis algoritm as linear complexity; it does a single pass witout any special cases or any additional grid. It advances automatically on watever side te next cell boundary is coming and andles any possible cases of nonmatcing grids at a one dimensional interface. 5 Numerical results On te unit square Ω = (, ) (, ) we consider te problem
7 y Nonconforming Grids wit Robin Interface Conditions 265 (Id )u(x, y) = x 3 (y 2 2) 6xy 2 + ( + x 2 + y 2 )sin(xy), (x, y) Ω, u = x 3 y 2 + sin(xy), (x, y) Ω, wose exact solution is u(x, y) = x 3 y 2 + sin(xy). We decompose te unit square into four nonoverlapping subdomains wit meses generated in an independent manner, as sown in Figure on te left. Te computed solution Initial Mes Computed solution x.8.6 y x Fig.. Initial mes and computed solution after two refinements. is te solution at convergence ( of te discrete algoritm (6)(7), wit) stopping criterion max,l/γ,l ((p Γ,l, + αu, ) ( p,l + αu,l ))ψ,l < 8, and α =. On Figure on te rigt, we sow te computed solution. Figure 2 on te left corresponds to te best approximation error of Teorem 4. On te rigt, we compare in te case of two subdomains te optimal log(h discrete error) log() 7 2 subdomains Nonconforming case alpa_opt 65 6 log(error) 2 number of iterations log() alpa Fig. 2. H error versus on te left and number of iterations versus α on te rigt. numerical α to te teoretical value, wic minimizes te convergence rate at te continuous level: α opt = [(π 2 π + )(( min ) 2 + )] 4. Te nonconforming meses ave 289 and 56 nodes respectively, and te discretization parame
8 266 Martin J. Gander, Caroline Japet, Yvon Maday, and Frédéric Nataf ters are =.65 and 2 =.32. We observe tat te optimal numerical α is very close to α opt. References Y. Acdou, C. Japet, Y. Maday, and F. Nataf. A new cement to glue nonconforming grids wit Robin interface conditions: te finite volume case. Numer. Mat., 92(4):593 62, 22. F. B. Belgacem and Y. Maday. Te mortar element metod for tree dimensional finite elements. RAIRO Matematical Modelling and Numerical Analysis, 3(2):289 32, 997. C. Bernardi, Y. Maday, and A. T. Patera. A new non conforming approac to domain decomposition: Te mortar element metod. In H. Brezis and J.L. Lions, editors, Collège de France Seminar. Pitman, 994. Tis paper appeared as a tecnical report about five years earlier. D. Braess and W. Damen. Stability estimates of te mortar finite element metod for 3dimensional problems. EastWest J. Numer. Mat., 6(4): , 998. P. Cevalier and F. Nataf. Symmetrized metod wit optimized secondorder conditions for te Helmoltz equation. In Domain decomposition metods, (Boulder, CO, 997), pages Amer. Mat. Soc., Providence, RI, 998. B. Després. Domain decomposition metod and te elmoltz problem. In SIAM, editor, Matematical and Numerical aspects of wave propagation penomena, pages Piladelpia PA, 99. M. J. Gander, L. Halpern, and F. Nataf. Optimal Scwarz waveform relaxation for te one dimensional wave equation. Tecnical Report 469, CMAP, Ecole Polytecnique, September 2. F. Gastaldi, L. Gastaldi, and A. Quarteroni. Adaptive domain decomposition metods for advection dominated equations. EastWest J. Numer. Mat., 4:65 26, 996. C. Japet. Optimized KrylovVentcell metod. Application to convectiondiffusion problems. In P. E. Bjørstad, M. S. Espedal, and D. E. Keyes, editors, Proceedings of te 9t international conference on domain decomposition metods, pages ddm.org, 998. C. Japet, Y. Maday, and F. Nataf. A new cement to glue nonconforming grids wit robin interface conditions: Te finite element case. to be submitted, 23. F. Nataf, F. Rogier, and E. de Sturler. Domain decomposition metods for fluid dynamics, NavierStoes equations and related nonlinear analysis. Edited by A. Sequeira, Plenum Press Corporation, pages , 995.
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