UNIFIED ERROR ANALYSIS

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1 UNIFIED ERROR ANALYSIS LONG CHEN CONTENTS 1. Lax Equivalence Teorem 1 2. Abstract error analysis 2 3. Application: Finite Difference Metod 3 4. Application: Finite Element Metod 4 5. Application: Conforming Discretization of Variational Problems 5 6. Application: Perturbed Discretization 6 7. Application: Nonconforming Finite Element Metods 8 8. Application: Finite Volume Metod 8 9. Application: Superconvergence of linear finite element metod 9 References 11 Let X, Y be two Banac spaces and L L (X, Y ) be a linear and bounded operator. We consider solving an operator equation: given y Y, find x X suc tat (1) Lx = y. We assume (1) is well defined, i.e., for any y Y, tere exists a unique solution to (1). Ten L is a one-to-one linear and bounded map and so is L 1 by te open mapping teorem. We sall study te convergence analysis in tis capter and refer to Capter: Inf-sup conditions for operator equations for well-posedness of (1). 1. LAX EQUIVALENCE THEOREM We are interested in te approximation of L. For H, 0, suppose L L (X, Y ) is a family of discretization of L. We consider te problem: given y Y, find x X, suc tat (2) L x = y. We also assume (2) is well defined. So L 1 is a linear and bounded map. But te norm L 1 could depend on te parameter. Te uniform boundedness will be called te stability of discretization. Namely, tere exists a constant C independent of suc tat (3) L 1 Y X C, for all H. Te consistency measures te approximation of L to L. We call te discretization is consistent if, for all x X, (4) Lx L x Y 0 as 0. Date: Oct, Last updated April 24,

2 2 LONG CHEN Te convergence of te discretization is defined as (5) x x X 0 as 0, were x and x are solutions to (2) and (1), respectively. We follow te book [4] to present te Lax equivalence teorem. Teorem 1.1 (Lax equivalence teorem). Suppose te discretization L of L is consistent, i.e., (4) olds for all x X, ten te stability (3) is equivalent to te convergence (5). Proof. We write te error as x x = L 1 (L x L x ) = L 1 (L x Lx). Here we use te fact L x = Lx = y. If te sceme is stable, ten x x X C Lx L x Y 0 as 0. On te oter and, for any y Y, let x = L 1 y and x = L 1 y. Te convergence x x means for any y Y L 1 y L 1 y as 0. By uniformly boundedness principle, we conclude L 1 is uniformly bounded. 2. ABSTRACT ERROR ANALYSIS Te Lax equivalence teorem may not be ready to use for te error analysis of discretizations since discrete spaces X, Y are used to approximate X, Y, respectively. Te equation may not be solved exactly, i.e., L x Lx. Instead, it solves approximately in Y, i.e., L x = y for some y Y. In some cases X X but in general X and X are different spaces so tat even x x does not make sense. Te norm X could be an approximation of X etc. We now refine te analysis to andle tese cases. Here we follow closely Temam [3]. Let us introduce two linear operators I : X X, and Π : Y Y. We consider te discrete problem: given y Y, find x X suc tat (6) L x = Π y in Y. Te consistency error is revised to and te stability is te same as before Π Lx L I x Y, L 1 Y X C, for all H. We sall prove if te sceme is consistent and stable, ten te discrete error I x x is convergent. Teorem 2.1. Suppose te sceme (L, Π ) is consistent wit order r, i.e., tere exists an operator I suc tat Π Lx L I x Y C r, and stable L 1 Y X C for all H, ten we ave te discrete convergence wit order r: I x x X C r.

3 UNIFIED ERROR ANALYSIS 3 Proof. Te proof is straigtforward using te definition, assumptions, and te identity: I x x = L 1 L (I x x ) = L 1 (L I x Π Lx). Remark 2.2. Te operator Π is in fact part of te discretization wile te operator I is introduced for te error analysis. For a given Π, te coices of I is not unique. If we really want to control x x, we need to define anoter linear operator P : X X to put a discrete function x into X in a stable way, i.e., P x X C x X. Ten by te triangle inequality, we ave x P x X x P I x X + P (I x x ) X x P I x X + C I x x X. Te additional error term x P I x X measures te approximation property of X by P X. Teorem 2.3. Suppose consistency: Π Lx L I x Y C 1 r, stability: L 1 Y X C 2 H, approximability: x P I x X C 3 s, ten we ave convergence x P x X C 2 (C 1 r + C 3 s ). Remark 2.4. Te domain of te operator L could be just a subspace of X. Strictly speaking we sould write L : dom(l) X Y. We leave a bigger space X in te analysis suc tat te embedding P : X X is easier to construct. And wit operators I and Π, te space X and Y do not ave to be Banac spaces. Under certain circumstance, e.g. wen L can be defined on X + X, we can break te consistence error Π Lx L I x into tree parts Π Lx Lx = L x Lx = y y: perturbation of te data Lx L x: consistence error L (x I x): approximation error. Te first term measures te approximation of te data in te subspace Y and te tird one measures te approximation of te function. Te middle one is te traditional consistence error introduced in Section 1. Te setting is 3. APPLICATION: FINITE DIFFERENCE METHOD (X, X ) = (Y, Y ) = (C( Ω),,Ω ); (X, X ) = (Y, Y ) = (R N, l ) I = Π as nodal interpolation; P : X X can be defined using bilinear finite element space on uniform grid; L = and L = is te 5-point stencil matrix. Note tat te domain of L is a subspace of X.

4 4 LONG CHEN Te consistency error is ( u) I u I l = max 1 i N u(v i) ( u I ) i C(u) 2, wic can be easily analyzed by te Taylor expansion. Here C(u) = d D4 i u,ω. Te stability 1 l l may be proved using te discrete maximum principle; see Capter: Finite Difference Metods. Te approximation property is given by te interpolation error estimate Here we use te same notation u I for P u I. u u I,Ω C 2 D 2 u,ω. 4. APPLICATION: FINITE ELEMENT METHOD Te setting is (X, X ) = (H0 1 (Ω), 1 ), and (Y, Y ) = (H 1 (Ω), 1 ); (X, X ) = (V, 1 ), and (Y, Y ) = (V, 1,). Here te dual norm f, v f 1, = sup, for f Y. v V v 1 I : H 1 0 (Ω) V is arbitrary; Π = Q : H 1 (Ω) = (H 1 0 (Ω)) V, i.e., Q f, v := f, v, v V. Wen Q is restricted to L 2 (Ω), it is te L 2 projection. Note tat (H 1 0 (Ω)) V, te operator Q can be also tougt as te natural embedding of te dual space and tus is omit in te notation sometimes. P : V H 1 0 (Ω) is te natural embedding since now V H 1 0 (Ω); L = : H 1 0 (Ω) H 1 (Ω) and L = Q LP : V V. Te equation is not solved exactly. It is solved in a weaker topology. Given a f H 1 (Ω), find u V suc tat L u = Q f in V. Te correct (comparing wit finite difference metod) stability for L 1 (7) u 1 Q f 1,, and can be proved as follows u 2 1 = L u, u = Q f, u Q f 1, u 1. is L 1 1, 1 : Comparing wit finite difference metod, now te consistency error is measured in a muc weaker norm. We could control te weaker norm Q f 1, by a stronger one f 1. Indeed implies (Q f, u ) = f, u f 1 u 1, (8) Q f 1, f 1. Let us denoted by v = I u. Using te fact L = Q LP and inequality (8), te consistency error is Q Lu L v 1, = Q L(u v ) 1, L(u v ) 1 u v 1.

5 UNIFIED ERROR ANALYSIS 5 By te stability result (7), we obtain u v 1 u v 1. Since v is arbitrary, by te triangle inequality, we obtain te Céa lemma (9) u u 1 2 inf v V u v 1. Te approximation property can be proved using Bramble-Hilbert lemma. For example, for linear finite element metod and u H 2 (Ω), inf u v 1 C u 2. v V Ten Teorem 2.3 will give optimal error estimate for finite element metods. Remark 4.1. Here we use only te stability and consistency. If we use te inner product structure and te ortogonality (or more general te variational formulation), we could improve te constant to 1 in (9). 5. APPLICATION: CONFORMING DISCRETIZATION OF VARIATIONAL PROBLEMS We now generalize te analysis for Poisson equation to general elliptic equations and sow te connection wit traditional error analysis of variational problems. Te operator L : V V is defined troug a bilinear form: for u, v V Lu, v := a(u, v). We consider te conforming discretization by coosing V V and te discrete operator L : V V V being te restriction of te bilinear form: for u, v V L u, v = a(u, v ). If we denoted by P : V V and Q : V V as te natural embedding. Ten by definition L = Q LP : V V. By defining in tis way, te operator L can be also tougt of as an extension of L. Te well known Lax-Milgram teorem (Lax again!) says if te bilinear form satisfies: coercivity: continuity: α u 2 V a(u, u). a(u, v) β u V v V. Ten tere exists a unique solution to te variational problem: given f V find u V suc tat (10) a(u, v) = f, v for all v V. Since V V, it also implies te existence and uniqueness of te solution u V (11) a(u, v ) = f, v for all v V. Tis establis te well posedness of (10) and (11). Te coercivity condition can be relaxed to te so-called inf-sup condition; see Capter: Inf-sup conditions for operator equations. Te continuity of te bilinear form a(, ) implies te continuity of L and L. Te coercivity can be used to prove te stability of L 1 in a straigtforward way: u 2 V a(u, u ) = f, u f V u V.

6 6 LONG CHEN For te conforming discretization, we ave te ortogonality (borrowed te name wen a(, ) is an inner product) i.e., a(u u, v ) = 0 for all v V. Lu = L u in V, wic can be also interpreted as a consistency result: Lu = f in V or in a less precise notation L u = Lu in V wit L u understood as Q Lu. Note tat te consistency is measured in te weak norm V but not in V. Te residual Lu L u 0 in V. In our setting, te consistence means: for any v V Lv = L v in V. In operator form, it is simply L = Q LP. Te traditional error analysis is as follows u u 2 V 1 α a(u u, u u ) (coercivity) = 1 α a(u u, u v ) (ortogonality) β α u u V u v V (continuity) wic implies te Céa lemma u u V β α inf u v V. v V Using our new framework of using stability and consistency, te estimate is like u v V 1 α L (u v ) V (stability) = 1 α Lu L v V (ortogonality) = 1 α L(u v ) V (consistency) β α u v V (continuity). Combined wit te triangle inequality, we get ( u u V 1 + β ) inf u v V. α v V 6. APPLICATION: PERTURBED DISCRETIZATION We first consider te quadrature of te rigt and side, i.e., f, v f, v. Te setting is as before except Π : V V defined as Π f, v = f, v. Note tat in tis case, we are solving te equation L u = Π f in V.

7 UNIFIED ERROR ANALYSIS 7 Te stability of L 1 is uncanged. We only need to estimate te consistency error Π L u L v V = Π f L v V. By te triangle inequality, we ave By te definition Π f L v V Q f L v V + Π f Q f V u v V + Π f Q f V. Π f Q f, w f, w f, w Π f Q f V = sup = sup. w V w w V w We end up wit te first Strang lemma u u V f, w f, w inf u v V + sup. v V w V w V Tat is, a perturbation of te data in V is included. Exercise 6.1. Use tree vertices quadrature rule, i.e. for a triangle τ formed by tree vertices x i, i = 1, 2, 3, g(x)dx 1 3 g(x i ) τ, 3 τ to compute te rigt and side (f, φ i ) and sow it can be simply written as Mf I, were M is a diagonal matrix. Prove optimal first order approximation in H 1 -norm for Poisson equation using tis quadrature. We ten include te approximation of te bilinear form wic includes te case of using numerical quadrature to compute te bilinear form a(, ) or a non-conforming discretization of L. We are solving te equation L u = Π f in V. To get similar estimate, we need to assume L is defined on V + V satisfying te stability and continuity wic migt be derived from te coercivity and continuity of te corresponding bilinear form ã (, ). Te space V is now endowed by a possibly different norm. Coercivity: for all v V L v, v α v 2. Continuity: for all u V + V, v V L u, v β u v. Ten it is straigtforward to prove te stability of L : (V, V ) (V, ). For te consistency error, Π f L v V Π f L u V + L (u v ) V f, w L u, w sup + u v. w V w Terefore we end up wit te second Strang lemma u u inf u v + C v V sup w V f, w L u, w. w

8 8 LONG CHEN 7. APPLICATION: NONCONFORMING FINITE ELEMENT METHODS 8. APPLICATION: FINITE VOLUME METHOD We consider te vertex-centered finite volume metod and refer to Capter: Finite Volume Metods for a detailed description and proof of corresponding results. Simply speaking, if we coose te dual mes by connecting a interior point to middle points of edges in eac triangle, te stiffness matrix is te same as tat from te linear finite element metod. So we sall use all te setting for te finite element metod. Te only difference is te rigt and side, i.e., Π : L 2 (Ω) V. Here we need to srink te space Y from H 1 (Ω) to L 2 (Ω). To define suc Π, let us introduce te piecewise constant space on te dual mes B and denoted by V 0,B. We rewrite te linear finite element space as V 1,T. Troug te point values at vertices, we define te following mapping N Π : V 1,T V 0,B as Π v = v (x i )χ bi. For any f L 2 (Ω), we ten define Π f V as Π f, v = (f, Π v ), for all v V. Te analysis below follows closely to Hackbusc [2]. Denoted by u G as te standard Galerkin (finite element) approximation and u B is te box (finite volume) approximation. Te equivalence of te stiffness matrices means Terefore by te stability of L 1, we ave By te definition L u G = Q f, L u B = Π f. u G u B 1 Q f Π f 1,. Q f Π f, v = (f, v Π v ). Denote te support te at basis function at x i as ω i. Note tat b i ω i and te operator I Π preserve constant function in te patc ω i and tus (f, v Π v ) bi f bi v Π v ωi C f bi v 1,ωi. Summing up and using Caucy Scwarz inequality, we get te first order convergence u G u B 1 C f. Furtermore, if te dual mes is symmetric in te sense tat we use barcenters as a vertex of control volumes. Ten we ave v = Π v, τ and tus, let f be te L 2 projection f to te piecewise constant function in eac triangle, (f, v Π v ) = (f f, v Π v ) f f v Π v V C 2 f 1 v 1. We ten obtain te second order convergence or te supercloseness of u G and ub : (12) u G u B 1 C 2 f 1. Wit suc relation, we can obtain optimal L 2 error estimate for u B and quasi-optimal L error estimate in two dimensions. Tat is u u B u u G + u G u B C 2 u 2 + u G u B 1 C 2 ( u 2 + f 1 ), τ

9 UNIFIED ERROR ANALYSIS 9 and similarly wit discrete embedding v C log v 1, u u B u u G + u G u B C 2 u 2 + C log u G u B 1 C log 2 ( u 2 + f 1 ). Remark 8.1. For non-symmetric dual mes but uniform triangulation, it is still possible to derive te super-closeness result using te tecnique in te next section. 9. APPLICATION: SUPERCONVERGENCE OF LINEAR FINITE ELEMENT METHOD In some cases we may ave a sarper estimate of te consistency error in te weak norm 1,. For example, for 1-D Poisson problem, using integration by parts and noting tat (u u I ) τ = 0, we get ((Q f L) u I, ϕ i ) = (u u I, ϕ i) = (u u I, ϕ i ) τ = 0, τ T wic implies te discrete error is zero. Namely u (x i ) = u(x i ) at eac grid points x i. On certain meses, we ave te following strengtened Caucy-Scwarz inequality (13) Q f L u I, v = ( u u I, v ) C 2 u 3 v 1. Ten consequently we obtain te superconvergence result u u I 1 Q f L u I 1, C 2 u 3. Rougly speaking, te symmetry of te mes will bring more cancelation wen measuring te consistency error. For example, te estimate (13) olds wen te two triangles saring an interior edge forms an O( 2 ) approximate parallelogram for most edges in te triangulation; See [1] for detailed condition on triangulation and a proof of (13). We sall give a simple proof of (13) on uniform grids by using te cancelation from te Taylor series. Here uniform grids we mean te tree directional triangulation obtained by using diagonals wit te same direction of uniform rectangular grids; see Fig. 1. FIGURE 1. A uniform mes for a square We assume tat te truncation error in l norm is of second order, i.e., ( u) I u I l C 2.

10 10 LONG CHEN Here u I is a vector and is te stencil matrix, c.f., te notation in te analysis of finite difference metod. We sall use te relation between finite element and finite difference metods, tat is, for uniform grids, we ave L u, v = u t A v = 2 u t v. Here we use boldface letters to denoted te vector formed by te function values at vertices. Namely for u V, we define u R N suc tat (u ) i = u (x i ). We scale te stiffness matrix A by 2 to get te five-point stencil. We coose I as te nodal interpolation and define Π : C 0 (Ω) V as Π f, v = 2 f t Iv = We divide te consistency error into two parts N 2 f(x i )v (x i ). Q f L u I 1, Q f Π f 1, + Π f L u I 1, = I 1 + I 2. To estimate I 1, we note Q f, v = N ( ω i fφ i )v (x i ) and 2 f(x i ) is a quadrature for te integral ω i fφ i. It is only exact for constant f restricted to one simplex. But for uniform grids, due to te configuration of ω i, te support of te at function φ i, te linear functional E i (f) = fφ i dx 2 f(x i ) ω i preserve linear polynomials in P 1 (ω i ). Namely E i (f ) = 0 for f P 1 (ω i ). Terefore E i (f) = E i (f f ) f f f 2,. Remark 9.1. For finite volume approximations, one can estimate te difference E i (f) = f dx 2 f(x i ). b i In tis case, b i is symmetric and b i = 2 and tus E i (f) can preserve linear polynomial in P 1 (b i ). Note tat for linear finite volume metods, te triangulation is more flexible. For triangulations consists of isosceles triangles, no matter wat te direction of te leg is, it results in te same discretization if we coose b i as type B control volume; see my notes on finite volume metods. Terefore we can estimate te I 1 as Q f Π f, v = Here we use estimate So we ave N E i (f)v (x i ) C f 2 N N 2 v (x i ) C v L 1 C v V C v 1. Q f Π f 1, C(f) 2. 2 v (x i ) C f 2 v 1.

11 UNIFIED ERROR ANALYSIS 11 To estimate I 2, by te consistency assumption, we ave Π f L u I, v = 2 (f I + u I ) v f I + u I C(u) 2 v 1. N 2 v (x i ) Terefore Π f L u I 1, C(u) 2. Here we consider Diriclet boundary condition and assume te truncation error for interior nodes is second order. For Neumann boundary condition or cell centered finite difference metods, te boundary or near boundary stencil is modified and te truncation error could be only first order O(); see Capter: Finite Difference Metods. We sall sow tat boundary stencil could be just first order wile te sceme is still second order in H 1 -norm. Suppose we label te boundary (or near boundary) nodes from 1 : N b and interior nodes from N b + 1 : N and te truncation error for boundary nodes is only O() and for interior nodes is still O( 2 ). Ten Π f L u I, v = 2 (f I + u I ) v f I + u I ( Nb = 2 v (x i ) + N i=n b +1 2 v (x i ) C(u) 2 ( ) v L1 ( Ω) + v L1 (Ω) C(u) 2 v 1. In te last step, we apply te following inequality v L1 ( Ω) v L2 ( Ω) v 1,Ω v 1. ) N 2 v (x i ) Te trick is to bound te boundary part by te L 1 norm of te trace and ten use te trace teorem to cange to H 1 -norm of te wole domain. In conclusion, we ave proved te superconvergence for finite difference metods (fivepoint stencil) or te linear finite element metod on uniform grids u I u 1 C(u, f) 2, by estimating te consistency error in a weaker norm. In tis approac we need stronger smootness assumption. For example, te constant C(u, f) = D 4 u + D 2 f. REFERENCES [1] R. E. Bank and J. Xu. Asymptotically exact a posteriori error estimators, Part I: Grids wit superconvergence. SIAM J. Numer. Anal., 41(6): , [2] W. Hackbusc. On first and second order box scemes. Computing, 41(4): , [3] R. Temam. Numerical Analysis. D. Reidel Publising Company, [4] G. Zang and Y. Lin. Functional analysis: I. Beijing University, 1987.