Some Error Estimates for the Finite Volume Element Method for a Parabolic Problem

Transcription

1 Computational Metods in Applied Matematics Vol. 13 (213), No. 3, pp c 213 Institute of Matematics, NAS of Belarus Doi: /cmam Some Error Estimates for te Finite Volume Element Metod for a Parabolic Problem Panagiotis Catzipantelidis Raytco Lazarov Vidar Tomée Abstract We study spatially semidiscrete and fully discrete finite volume element metods for te omogeneous eat equation wit omogeneous Diriclet boundary conditions and derive error estimates for smoot and nonsmoot initial data. We sow tat te results of our earlier work [Mat. Comp. 81 (212), 1 2] for te lumped mass metod carry over to te present situation. In particular, in order for error estimates for initial data only in L 2 to be of optimal second order for positive time, a special condition is required, wic is satisfied for symmetric triangulations. Witout any suc condition, only first order convergence can be sown, wic is illustrated by a counterexample. Improvements old for triangulations tat are almost symmetric and piecewise almost symmetric. 21 Matematical subject classification: 65M6, 65M15. Keywords: Finite Volume Metod, Parabolic Partial Differential Equations, Nonsmoot Initial Data, Error Estimates. 1. Introduction We consider te model initial-boundary value problem u t u =, in Ω, u =, on Ω, for t, wit u() = v, in Ω, (1.1) were Ω is a bounded convex polygonal domain in R 2. We restrict ourselves to te omogeneous eat equation, tus witout a forcing term, so tat te initial values v are te only data of te problem. Tis problem as a unique solution u(t), under appropriate assumptions on v, and tis solution is smoot for t >, even if v is not. To express te smootness properties of te solution of (1.1), let, for q, Ḣ q L 2 (Ω) be te Hilbert space defined by te norm ( ) 1/2, w q = λ q j (w, φ j) 2 were (w, ϕ) = wϕ dx, (1.2) j=1 Panagiotis Catzipantelidis Department of Matematics, University of Crete, 7149 Heraklion, Greece catzipa@mat.uoc.gr. Raytco Lazarov Department of Matematics, Texas A&M University, College Station, TX 77843, USA; and Institute of Matematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Boncev str., bl. 8, 1113 Sofia, Bulgaria lazarov@mat.tamu.edu. Vidar Tomée Matematical Sciences, Calmers University of Tecnology and te University of Gotenburg, SE Göteborg, Sweden; and Institute of Applied and Computational Matematics, FORTH, 7111 Heraklion, Greece tomee@calmers.se. Ω Unautenticated

2 252 Panagiotis Catzipantelidis, Raytco Lazarov, Vidar Tomée and were {λ j } j=1, {φ j } j=1 are te eigenvalues, in increasing order, and ortonormal eigenfunctions of in Ω, wit omogeneous Diriclet boundary conditions on Ω. Tus w = w = (w, w) 1/2 is te norm in L 2 = L 2 (Ω), w 1 = w te norm in H 1 = H(Ω) 1 and w 2 = w is equivalent to te norm in H 2 (Ω) wen w = on Ω. Eigenfunction expansion and Parseval s relation sow for te solution u(t) = E(t)v of (1.1) te stability and smooting estimate E(t)v p Ct (p q)/2 v q, for q p, and t >. (1.3) In fact, since te smallest eigenvalue is positive, a factor of e ct, wit c >, may be included in te rigt-and side, and tis olds for all our stability, smooting and error estimates trougout our paper. Since our interest ere is in small time we sall not keep track of tis decay for large time below. We sall also use te norm w C k = γ k sup x Ω D γ xw(x) in C k = C k ( Ω), wit C = C, te space of continuous functions on Ω. Here for γ = (γ 1, γ 2 ), D γ x = ( / x 1 ) γ 1 ( / x 2 ) γ 2 and γ = γ 1 + γ 2. We first recall some facts about te spatially semidiscrete standard Galerkin finite element metod for (1.1) in te space of piecewise linear functions S = {χ C : χ τ linear, τ T ; χ Ω = }, were {T } is a family of regular triangulations T = {τ} of Ω, wit denoting te maximum diameter of te triangles τ T. Tis metod defines an approximation u (t) S of u(t), for t, from (u,t, χ) + ( u, χ) =, χ S, for t, wit u () = v, (1.4) were v S is an approximation of v. It is well known tat we ave te smoot data error estimate, valid uniformly down to t =, see, e.g., [12], u (t) u(t) C 2 v 2, if v v C 2 v 2, for t. (1.5) We also ave a nonsmoot data error estimate, for v only assumed to be in L 2, wic is of optimal order O( 2 ) for t bounded away from zero, but deteriorates as t, u (t) u(t) C 2 t 1 v, if v = P v, for t >, (1.6) were P denotes te ortogonal L 2 -projection onto S. Note tat te coice of discrete initial data is not as general in tis case as in (1.5). We empasize tat te triangulations T are assumed to be independent of t, and tus te use of finer T for t small is not considered ere. We note tat a possible coice in (1.5) is v = P v, and ence, by interpolation, we ave te intermediate result between (1.5) and (1.6), u (t) u(t) C 2 t 1/2 v 1, if v = P v, for t >. (1.7) Recently, in [4], we sowed results similar to (1.5) (1.7) for te lumped mass finite element metod, wic may be defined by replacing te L 2 -inner product in te first term in (1.4) by te quadrature approximation (u,t, χ), were, wit I : C S being te interpolant defined by I v(z) = v(z) for any vertex z of T, (χ, ψ) = I (χ ψ) dx, χ, ψ S. Ω Unautenticated

3 Error Estimates for te Finite Volume Element Metod 253 z V z z τ τ z z Figure 1. Left: A union of triangles tat ave a common vertex z; te dotted line sows te boundary of te corresponding control volume V z. Rigt: A triangle τ partitioned into te tree subregions τ z. Improving earlier results, we demonstrated tat (1.5) remains valid for te lumped mass metod, but tat (1.6) requires restrictive conditions on {T }, caused by te use of quadrature in (1.4), and satisfied, in particular, for symmetric triangulations. We remark tat te coice of discrete initial data in te analogue of (1.7) was incorrectly stated in [4], see Section 3 below. In te present paper our purpose is to carry over te analysis in [4] to te finite volume element metod for problem (1.1). Tis metod is based on a local conservation property associated wit te differential equation. Namely, integrating (1.1) over any region V Ω and using Green s formula, we obtain u t dx u n dσ =, for t, (1.8) V V were n denotes te unit exterior normal vector to V. Te semidiscrete finite volume element approximation ũ (t) S will satisfy (1.8) for V in a finite collection of subregions of Ω called control volumes, te number of wic will be equal to te dimension of te finite element space S. Tese control volumes are constructed in te following way. Let z τ be te barycenter of τ T. We connect z τ by line segments to te midpoints of te edges of τ, tus partitioning τ into tree quadrilaterals τ z, z Z (τ), were Z (τ) are te vertices of τ. Ten wit eac vertex z Z = τ T Z (τ) we associate a control volume V z, wic consists of te union of te subregions τ z, saring te vertex z (see Figure 1, left). We denote te set of interior vertices of Z by Z. Te semidiscrete finite volume element metod for (1.1) is ten to find ũ (t) S suc tat ũ,t dx V z ũ n dσ =, V z z Z, for t, wit ũ () = v, (1.9) were v S is an approximation of v. Tis problem may also be expressed in a weak form. For tis purpose we introduce te finite-dimensional space of piecewise constant functions Y = {η L 2 : η Vz = constant, z Z ; η Vz =, z Z \ Z }. We now multiply (1.9) by η(z) for an arbitrary η Y, and sum over z Z to obtain te Petrov Galerkin formulation (ũ,t, η) + a (ũ, η) =, η Y, for t, wit ũ () = v, (1.1) Unautenticated

4 254 Panagiotis Catzipantelidis, Raytco Lazarov, Vidar Tomée were te bilinear form a (, ) : S Y R is defined by a (χ, η) = η(z) χ n dσ, χ S, η Y. (1.11) z Z V z Obviously, we can define a (, ) also for χ replaced by w H 2, and using Green s formula we ten easily see tat a (w, η) = ( w, η), w H 2, η Y. We sall now rewrite te Petrov Galerkin metod (1.1) as a Galerkin metod in S. For tis purpose, we introduce te interpolation operator J : C Y by J u = z Z u(z)ψ z, were Ψ z is te caracteristic function of te control volume V z. It is known tat J is selfadjoint and positive definite, see [5], and ence te following defines an inner product, on S, χ, ψ = (χ, J ψ), χ, ψ S. (1.12) Also, te corresponding discrete norm is equivalent to te L 2 -norm, uniformly in, i.e., wit C c >, c χ χ C χ, χ S, were χ χ, χ 1/2, see [5]. Furter, in [2], it is sown tat a (χ, J ψ) = ( χ, ψ), χ, ψ S, and terefore, a (, ) is symmetric and a (χ, J χ) = χ 2, for χ S. Wit tis notation, (1.1) may equivalently be written in Galerkin form as ũ,t, χ + ( ũ, χ) =, χ S, for t, wit ũ () = v. (1.13) Our aim is tus to sow analogues of (1.5) (1.7) for te solution of (1.13), wit te appropriate coices of v, i.e., ũ (t) u(t) C 2 t 1+q/2 v q, for t >, q =, 1, 2. (1.14) Tis will be done below for q = 2, and in te case q = 1 under te additional assumption tat {T } is quasi-uniform. However, for q =, as in [4], we are only able to sow (1.14) under an additional ypotesis, expressed in terms of te quadrature error operator Q : S S, defined by ( Q ψ, χ) = ε (ψ, χ), χ, ψ S, (1.15) were ε (, ) is te quadrature error defined ere by ε (f, χ) = (f, J χ) (f, χ), f L 2, χ S, (1.16) and requiring Q ψ C 2 ψ, ψ S. (1.17) Unautenticated

5 Error Estimates for te Finite Volume Element Metod 255 We will sow tat tis assumption is satisfied for symmetric triangulations T. Symmetry of T, owever, is a severe restriction wic can only old for special sapes of Ω. For tis reason we will also consider less restrictive families {T }. We will demonstrate tat (1.17) olds for almost symmetric families (discussed in Section 4), wit te addition of a logaritmic factor; we also sow tat tis logaritmic factor is not needed in one space dimension. Furter, for piecewise almost symmetric families of triangulations, see Section 4, te inequality (1.17) olds wit an O( 3/2 ) bound. We ten give two examples of nonsymmetric triangulations suc tat (1.14) does not old for q =. In te first example we construct {T } suc tat te convergence factor is at most of order O() for t >, and in te second example, wit nonsymmetry only along a line, of order O( 3/2 ). Witout any additional condition on T we are only able to sow te nonoptimal order error estimate ũ (t) u(t) Ct 1/2 v, if v = P v, for t >. We remark tat in [11], in te more general case of a parabolic integro-differential equation, te nonsmoot data error estimate (1.14), for q =, wit an extra factor log, was stated for any quasi-uniform family {T }. Unfortunately, tis result is in contradiction to our above counterexamples, and its proof incorrect. We also discuss optimal order O() error estimates for te gradient of ũ u, under various assumptions on te smootness of v and coices of v. Furter, in a separate section, we consider briefly te extension of our results for te spatially semidiscrete problem to te fully discrete backward Euler and Crank Nicolson finite volume metods. As for te lumped mass metod in [4], our analysis yields improvements of earlier results, in [3], were it was sown tat, for smoot initial data and v = R v, and ũ (t) u(t) C 2 v 3, for t >, (ũ (t) u(t)) Cɛ 1 v 2+ɛ, for t >, ɛ > small. As in te case of te lumped mass metod in [4], tese improvements are made possible by combining te error estimates (1.5) (1.7) for te standard Galerkin finite element metod wit bounds for te difference δ = ũ u, wic, by (1.13) and (1.4), satisfies δ t, χ + ( δ, χ) = ε (u,t, χ), χ S, for t. (1.18) In te final section we sketc te extension of te teory developed above to more general parabolic equations, considering te initial-boundary value problem u t + Au =, in Ω, u =, on Ω, for t, wit u() = v, in Ω, (1.19) were Au = (α u) + βu, wit α a smoot symmetric, positive definite 2 2 matrix function on Ω and β a non-negative smoot function. Here, let u (t) S denote te standard Galerkin finite element approximation of u(t), defined by (u,t, χ) + a(u, χ) =, χ S, for t, wit u () = v, (1.2) were v S is an approximation of v and a(w, ϕ) = (α w, ϕ) + (βw, ϕ), for w, ϕ H 1. (1.21) In a straigtforward way te estimates (1.5) (1.7) extend to te solution of (1.2). Unautenticated

6 256 Panagiotis Catzipantelidis, Raytco Lazarov, Vidar Tomée Te natural generalization of te finite volume metod (1.1) would now be to find ũ (t) S suc tat ũ,t, χ + a (ũ, J χ) =, χ S, for t, wit ũ () = v, (1.22) were, instead of (1.11), one uses te bilinear defined by a (ψ, η) = z Z ( ) η(z) (α ψ) n dσ + βψ dx, ψ S, η Y. (1.23) V z V z It is known tat, in general, te bilinear form a (ψ, J χ) is nonsymmetric on S but it is not far from being symmetric, or a (χ, J ψ) a (ψ, J χ) C χ ψ, cf. [5]. Also, if α and β are constants over eac τ T, ten, see, e.g., [2, 6], a (ψ, J χ) = (α ψ, χ) + (βψ, J χ), ψ, χ S, (1.24) and tus a (ψ, J χ) is symmetric, since as we sall sow (βψ, J χ) = (βχ, J ψ). Terefore, since symmetry is important in our analysis, we introduce te modified bilinear form ã (ψ, η) = ( ) η(z) ( α ψ) n dσ + βψ dx, ψ S, η Y, (1.25) z Z V z V z were, for z τ, τ T, α(z) = α(z τ ) and β(z) = β(z τ ), wit z τ te barycenter of τ. Tis coice of ã (, ) leads to te finite volume element metod, to find ũ (t) S suc tat ũ,t, χ + ã (ũ, J χ) =, χ S, for t, wit ũ () = v, (1.26) and for tis te desired analogues of te estimates (1.14) are establised in Teorems Te following is an outline of te paper. In Section 2, we introduce notation and give some preliminary material needed for te analysis of te finite volume element metod. Furter, we derive smoot and nonsmoot initial data estimates for te gradient of te error in te standard Galerkin metod. In Section 3 we derive te error estimates (1.14) discussed above under te different assumptions on smootness of data and te triangulations {T }. In Section 4 we sow tat assumption (1.17) is valid for symmetric meses, and discuss te corresponding properties for almost symmetric and piecewise almost symmetric meses. In Section 5 we present two nonsymmetric triangulations in two space dimensions for wic optimal order L 2 -convergence for nonsmoot data does not old. In Section 6 we consider briefly te application to te fully discrete backward Euler and Crank Nicolson finite volume metods. Finally, Section 7 contains te extension of Section 3 to more general parabolic equations. 2. Preliminaries In tis section we sow a smooting property for te finite volume element metod, and discuss te quadrature associated wit tis metod. We also derive some estimates for te gradient of te error in te standard Galerkin finite element metod wic will be needed later. Unautenticated

7 Error Estimates for te Finite Volume Element Metod 257 We first recall tat for te standard Galerkin metod, one may introduce te discrete Laplacian : S S by and write te problem (1.4) as ( ψ, χ) = ( ψ, χ), ψ, χ S, u,t u =, for t, wit u () = v. (2.1) Let {λ j } N j=1 denote te eigenvalues, in increasing order, and {φ j } N j=1 te corresponding eigenfunctions of, ortonormal wit respect to (, ), were N = dim S. Ten we ave for te solution operator E (t) = e t of (2.1), by eigenfunction expansion, N u (t) = E (t)v = e λ j t (v, φ j )φ j, for t. j=1 Te following smooting property analogous to (1.3) olds for v S and t >, p D l te (t)v Ct l (p q)/2 q v, l, p, q =, 1, 2l + p q, (2.2) wit D t = / t. Turning to te finite volume metod (1.13), we now introduce te discrete Laplacian : S S, corresponding to te inner product, in (1.12), by ψ, χ = ( ψ, χ), ψ, χ S. (2.3) Te finite volume metod (1.13) can ten be written in operator form as ũ,t ũ =, for t, wit ũ () = v. (2.4) For te solution operator Ẽ(t) = e t of (2.4) we ave ũ (t) = Ẽ(t)v N = e λ j t v, φ j φ j, for t, (2.5) j=1 were { λ j } N j=1 and { φ j } N j=1 are te eigenvalues, in increasing order, and te corresponding eigenfunctions, ortonormal wit respect to,, of te positive definite operator. For Ẽ (t) te following analogue of (2.2) olds, cf. [4, Lemma 2.1]. Lemma 2.1. For Ẽ defined by (2.5) we ave, for v S and t >, p D l tẽ(t)v Ct l (p q)/2 q v, l, p, q =, 1, 2l + p q. Proof. Introducing te square root G = ( ) 1/2 : S S of, we get v 2 = ( N )v, v = λ j v, φ j 2 = G v 2. Since te norms and are equivalent on S, we find, for t >, p DtẼ(t)v l 2 C G N p Dl tẽ(t)v 2 = C ( λ j ) 2l+p q e 2 λ j t ( λ j ) q v, φ j 2 j=1 j=1 C t (2l+p q) G q v 2 C t (2l+p q) q v 2. Unautenticated

8 258 Panagiotis Catzipantelidis, Raytco Lazarov, Vidar Tomée Te quadrature error functional ε (, ) defined by (1.16) as an important role in our analysis below. For tis reason we recall te following lemma, cf. [3]. Lemma 2.2. For te error functional ε, defined by (1.16), we ave ε (f, ψ) C p+q p f q ψ, f H 1, ψ S, and p, q =, 1. Proof. Since τ (J ψ ψ) dx = for ψ linear in τ, for any τ T, see [5], we ave tat J ψ ψ is ortogonal to S, te set of piecewise constants on T. Hence ε (f, ψ) = (f, J ψ ψ) = (f P f, J ψ ψ), were P is te ortogonal projection onto S. Te lemma now easily follows since we ave J ψ ψ C ψ and P f f C f. Te following estimate olds for te quadrature error operator Q in (1.15). Lemma 2.3. Let and Q be te operators defined by (2.3) and (1.15). Ten Q χ + Q χ C p+1 p χ, χ S, p =, 1. Proof. By (1.15) and Lemma 2.2, wit ψ = Q χ and q = 1, it follows easily tat Q χ 2 = ε (χ, Q χ) C p+1 p χ Q χ, for p =, 1, wic sows te desired estimate for Q χ. Also, by te definition of, Lemma 2.2 wit q = sows, for p =, 1, Q χ 2 = ( Q χ, Q χ) = ε (χ, Q χ) C p p χ Q χ. Since te norms and are equivalent on S, tis implies te bound for te remaining term Q χ. In addition to te ortogonal L 2 -projection P, our error analysis will use te Ritz projection R : H 1 S defined by It is well known tat R satisfies ( R w, χ) = ( w, χ), χ S. R w w + (R w w) C q w q, for w Ḣq, q = 1, 2. (2.6) We close wit some estimates for te gradient of te error, sligtly generalizing tose of [4, Teorem 2.1]. Teorem 2.1. Let u and u be te solutions of (1.1) and (2.1). Ten, for t >, C v 2, if (v v) C v 2, (u (t) u(t)) Ct 1/2 v 1, if v v C v 1, Ct 1 v, if v = P v. Proof. In [4, Teorem 2.1] tis was sown wit v = R v in te first two estimates, and tus it remains to bound E (t)(v R v). Wit ϑ := v R v we find easily, by Lemma 2.1, for smoot data, E (t)ϑ() ϑ() C v 2, and for mildly nonsmoot data, E (t)ϑ() Ct 1/2 ϑ() Ct 1/2 v 1. Unautenticated

9 Error Estimates for te Finite Volume Element Metod Smoot and Nonsmoot Initial Data Error Estimates In tis section we derive optimal order error estimates for te finite volume element metod (1.13), wit initial data v in Ḣ2, Ḣ 1 and L 2. For v Ḣ2, te error estimate is te same as tat for te standard Galerkin finite element metod, and tis is also te case for v Ḣ1, provided te family of finite element spaces is quasi-uniform. In te case v L 2, wit discrete initial data v = P v, in order to derive an optimal order estimate analogous to (1.6), we need to impose condition (1.17) for te quadrature error operator Q. In Section 4 we verify tis condition for symmetric meses. In te general case we are only able to sow a nonoptimal order O() error bound in L 2, wereas for te gradient of te error an optimal order O() bound still olds. Te estimates and teir proofs are analogous to tose for te lumped mass metod derived in [4], since te operators Ẽ, and Q, defined in Section 2, ave properties similar to tose of te corresponding operators for te lumped mass metod. References to [4] will terefore be given in some of te proofs below. We begin wit smoot initial data, v Ḣ2. Teorem 3.1. Let u and ũ be te solutions of (1.1) and (2.4). Ten ũ (t) u(t) C 2 v 2, if v v C 2 v 2, for t. Proof. Since, by (1.5), te corresponding error bound olds for te solution u of te standard Galerkin metod, it suffices to consider te difference δ = ũ u. Also, by te stability estimates of Lemma 2.1, we may assume tat v = R v. By te definition (1.15) of Q, δ satisfies (1.18), and ence δ t δ = Q u,t, for t, wit δ() =, (3.1) were u is te solution of (1.4). By Duamel s principle tis sows δ(t) = t Ẽ (t s) Q u,t (s) ds. (3.2) Using te fact tat Ẽ(t) = D t Ẽ (t), and Lemmas 2.1 and 2.3, we easily get and ence Ẽ(t) Q χ Ct 1/2 Q χ C 2 t 1/2 χ, for χ S, (3.3) t δ(t) C 2 (t s) 1/2 u,t (s) ds. Here, since R = P, we obtain, by first applying Lemma 2.1, and ence u,t (s) Cs 1/2 u,t () = Cs 1/2 R v Cs 1/2 v = Cs 1/2 v 2, wic completes te proof. t δ(t) C 2 (t s) 1/2 s 1/2 ds v 2 = C 2 v 2, Unautenticated

10 26 Panagiotis Catzipantelidis, Raytco Lazarov, Vidar Tomée We now consider mildly nonsmoot initial data, v Ḣ1. Here we sall need to assume te stability of P in Ḣ1, or P w C w 1, wic does not old for arbitrary families of triangulations. However, a sufficient condition for suc stability of P is te global quasiuniformity of {T }. Indeed, tis assumption implies te inverse inequality χ C 1 χ, wic combined wit te error bound R w w C w 1 sows te desired stability of P. Teorem 3.2. Let u and ũ be te solutions of (1.1) and (2.4). Ten for t > ũ (t) u(t) C 2 t 1/2 v 1, if v = P v and P v C v 1. Proof. Since by (1.7), te corresponding error estimate olds for te solution u of te standard Galerkin metod (witout te condition on P ), it suffices as above to bound δ = ũ u. We use (3.2) to write { t/2 δ(t) = t + }Ẽ (t s) Q u,t (s) ds = δ 1 (t) + δ 2 (t). (3.4) t/2 Using again (3.3), we ave, since u,t (s) Cs 1 P v Cs 1 v 1, tat Integrating by parts, we obtain t δ 2 (t) C 2 (t s) 1/2 u,t (s) ds C 2 t 1/2 v 1. t/2 δ 1 (t) = [ Ẽ (t s) Q u (s) ] t/2 t/2 D s Ẽ (t s) Q u (s) ds. (3.5) Employing (3.3), Lemmas 2.1 and 2.3 we now find, similarly to te above, δ 1 (t) C 2 t 1/2( u (t/2) + P v ) t/2 + C 2 (t s) 3/2 u (s) ds C 2 t 1/2 v 1. Togeter tese estimates complete te proof. Te analogous result and its proof also old for te lumped mass metod, wic sould replace te case q = 1 in [4, Teorem 3.1], since (1.7) does not old for v = R v. Next, we turn to te nonsmoot initial data error estimate. Teorem 3.3. Let u and ũ be te solutions of (1.1) and (2.4). If (1.17) olds and v = P v, ten ũ (t) u(t) C 2 t 1 v, for t >. Proof. Tis follows easily from te fact tat for Q satisfying (1.17) we ave, Ẽ(t) Q P v Ct 1 Q P v C 2 t 1 v, for t >. (3.6) Tis inequality is te necessary and sufficient condition for te desired bound to old by te following lemma, wic is proved in te same way as [4, Teorem 4.1]. Unautenticated

11 Error Estimates for te Finite Volume Element Metod 261 Lemma 3.1. Let u and ũ be te solutions of (1.1) and (2.4). Ten ũ (t) u(t) + Ẽ(t) Q v C 2 t 1 v, if v = P v, for t >. Condition (1.17) will be discussed in more detail in Section 4 below. Lemma 2.3, witout additional assumptions on te mes, we ave Note tat, by Q χ C Q χ C χ, χ S, and tat te lower order error estimate of te following teorem always olds. Te proof is te same as tat of [4, Teorem 4.3]. We sall sow in Section 5 tat a O() bound is te best possible for general triangulation families {T }. Teorem 3.4. Let u and ũ be te solutions of (1.1) and (2.4). Ten ũ (t) u(t) Ct 1/2 v, if v = P v, for t >. We end tis section by stating optimal order estimates for te gradient of te error. Note tat no additional assumption on {T } is required. Teorem 3.5. Let u and ũ be te solutions of (1.1) and (2.4). Ten, for t >, C v 2, if (v v) C v 2, (ũ (t) u(t)) Ct 1/2 v 1, if v v C v 1, Ct 1 v, if v = P v. Proof. For te first two estimates it suffices, by te stability and smootness estimates of Lemma 2.1, to consider v = R v. For tis coice of te initial data te proofs are identical to tose in [4, Teorem 3.1]. In te nonsmoot data case, te proof is te same as tat of [4, Teorem 4.4]. 4. Symmetric and Almost Symmetric Triangulations In tis section we first sow tat for families of triangulations {T } tat are symmetric, in a sense to be defined below, assumption (1.17) is satisfied and terefore, by Teorem 3.3, te optimal order nonsmoot data error estimate olds. We sall ten relax te symmetry requirements and consider almost symmetric families of triangulations, consisting of O( 2 ) perturbations of symmetric triangulations. In tis case we sow tat (1.17) is satisfied wit an additional logaritmic factor and, as a consequence, an almost optimal order nonsmoot data error estimate olds. Finally for te less restrictive class of piecewise almost symmetric families {T } we derive a O( 3/2 ) order nonsmoot data error estimate. In addition to te quadrature error operator Q defined in (1.16) we sall work wit te symmetric operator M : S S, defined by were we use te inner product ε (ψ, χ) = [ψ, M χ], ψ, χ S, (4.1) [ψ, χ] = z Z ψ(z)χ(z), ψ, χ S. (4.2) Unautenticated

12 262 Panagiotis Catzipantelidis, Raytco Lazarov, Vidar Tomée ν τ = ν τ 3 ζ 2 ζ 3 τ 2 τ 1 ζ 1 τ τ 3 Π ζ τ 4 ζ τ 5 τ 6 ζ 6 ζ 4 ν τ 1 = ν τ 4 ν τ 2 ζ 5 Figure 2. Left: A triangle τ. Rigt: A patc Π ζ around a vertex ζ. To determine te form of tis operator, we introduce some notation. For z Z an interior vertex of T, we define te patc Π z = { τ : τ T, z τ}, were for simplicity we ave assumed tat τ = τ. Furter, for z a vertex of τ T, we denote by z+ τ and z τ te oter two vertices of τ. We ten define M Πz χ := 1 τ ( χ(z 54 +) τ 2χ(z) + χ(z ) ) τ, (4.3) τ Π z for wic te following olds. Lemma 4.1. For te operator M defined by (4.1) we ave, for z Z, M χ(z) = M Πz Proof. In view of (1.16), we may write ε (ψ, χ) = (ψ, J χ) (ψ, χ) = τ T Πz χ wit M χ given by (4.3). (4.4) τ (ψj χ ψχ) dx. (4.5) For τ T we denote its vertices by ν τ 1, ν τ 2, ν τ 3 and set ν τ 4 = ν τ 1, ν τ = ν τ 3, see Figure 2. Writing w j = w(ν τ j ) for a function w on τ, we obtain, after simple calculations, and Tus τ τ ψj χ dx = τ 18 τ ψχ dx = τ 12 3 ψ j (22χ j + 7χ j 1 + 7χ j+1 ), (4.6) j=1 3 ψ j (2χ j + χ j 1 + χ j+1 ). j=1 (ψj χ ψ χ) dx = τ 54 Summation over τ T, (4.1) and (4.5) sow [ψ, M χ] = z Z Tis implies (4.4) and tus completes te proof. 3 ψ j (χ j+1 2χ j + χ j 1 ). j=1 ψ(z)m Πz χ, ψ, χ S. Unautenticated

13 Error Estimates for te Finite Volume Element Metod 263 z z z Figure 3. Patces wic are symmetric wit respect to te vertex z. We say tat T is symmetric at z Z, if te corresponding patc Π z is symmetric around z, in te sense tat if x Π z, ten z (x z) = 2z x Π z. We say tat T is symmetric if it is symmetric at eac z Z. Te patc Π ζ in Figure 2 is nonsymmetric wit respect to ζ, wereas triangulations wic are built up of eiter of te patces sown in Figure 3 are symmetric. Symmetric triangulations exist only for special domains, suc as parallelograms, but not for general polygonal domains. We now sow te sufficiency of symmetry of {T } for condition (1.17) for te operator Q, and ence, by Teorem 3.3, for te nonsmoot data error estimate. Teorem 4.1. If te family {T } is symmetric, ten (1.17) olds. Proof. Te proof, by duality, follows tat of [4, Teorem 5.1]. For given χ S we define ϕ = ϕ χ Ḣ1 as te solution of te Diriclet problem ϕ = χ in Ω, ϕ = on Ω. Since Ω is convex, we ave ϕ Ḣ2 and ϕ 2 C χ. Wit I te finite element interpolation operator into S, we ave, for any ψ S, (Q ψ, χ) Q ψ = sup χ S χ ( Q ψ, ϕ) = sup χ S χ ( Q ψ, (ϕ I ϕ)) ( Q ψ, I ϕ) sup + sup χ S χ χ S χ By te obvious error estimate for I and Lemma 2.3, wit p =, we find Q ψ ϕ 2 I C sup χ S χ = I + II. (4.7) C 2 ψ. (4.8) To estimate II, we employ (1.15) and (4.1) to rewrite te numerator in te form ( Q ψ, I ϕ) = ε (ψ, I ϕ) = [ψ, M I ϕ]. (4.9) To bound M I ϕ, we consider an arbitrary vertex z = ζ Z. Let Π ζ be te corresponding patc of T, wit vertices {ζ j } K j=1, numbered counter-clockwise, wit ζ j+k = ζ j for all j. Also denote by {τ j } K j=1 te triangles of T in Π ζ, wit τ j aving vertices ζ, ζ j, ζ j+1, and set τ = τ K (see Figure 2). Ten Lemma 4.1 implies M I ϕ(ζ ) = M Π ζ I ϕ = 1 54 K ω j (ϕ(ζ j ) ϕ(ζ )), (4.1) wit ω j = τ j 1 + τ j. By assumption, te patc Π ζ is symmetric and ence, by (4.1), we can express M I ϕ(ζ ) as a linear combination of terms of te form ϕ(ζ j ) 2ϕ(ζ ) + ϕ(ζ j), Unautenticated j=1

14 264 Panagiotis Catzipantelidis, Raytco Lazarov, Vidar Tomée Figure 4. Left: An almost symmetric triangulation. Rigt: A piecewise almost symmetric triangulation. were ζ is te midpoint of te vertices ζ j and ζ j of Π ζ. Hence M I ϕ(ζ ) = for ϕ linear in Π ζ and, as in [4], we may apply te Bramble Hilbert lemma to obtain M I ϕ(ζ ) C 2 Π ζ 1/2 ϕ H 2 (Π ζ ) C 3 ϕ H 2 (Π ζ ). (4.11) Employing tis estimate for all patces Π z of T, we obtain, for any ψ S, [ψ, M I ϕ] C 3 ψ(z) ϕ H 2 (Π z) C 2 ψ ϕ 2 C 2 ψ χ. (4.12) z Z Hence, in view of (4.7) and (4.9), we obtain II C 2 ψ. completes te proof. Togeter wit (4.8) tis We now want to sligtly weaken te assumption about symmetry. We say tat a family of triangulations {T } is almost symmetric if eac T is a perturbation by O( 2 ) of a symmetric triangulation, uniformly in, in te sense tat wit eac patc Π z of T tere is an associated symmetric patc from wic Π z is obtained by moving eac of its vertices by O( 2 ). Suc triangulations exist for any convex quadrilateral, cf. Figure 4. We note tat various special triangulations ave been used in te past for obtaining iger order accuracy for te gradient of te finite element solution (super-convergent rates of O( 2 ) or O( 2 l )), see, e.g., [7, 1, 13]. For example, te strongly regular triangulations from [1], requiring tat any two adjacent triangles form almost a parallelogram (a deviation of a parallelogram by O( 2 )), are almost symmetric meses in our terminology. We sall sow tat, in tis case, we ave almost optimal order convergence for nonsmoot initial data. Teorem 4.2. If te family {T } is almost symmetric, ten Q ψ C 2 l 1/2 ψ, ψ S, were l = 1 + log. (4.13) Hence, for te solution of (1.13), wit v = P v, we ave ũ (t) u(t) C 2 l 1/2 t 1 v, for t >. (4.14) In te proof we sall need te following Sobolev type inequality, were te H k denote seminorms wit only te derivatives of igest order k. Lemma 4.2. Let B be a fixed bounded domain, satisfying te cone property. Ten we ave, for < ɛ < 1, ϕ(z ) ϕ(z) sup Cɛ 1/2( ) ϕ z,z B, z z z z 1 ɛ H 1 (B) + ϕ H 2 (B), ϕ H 2 (B). Unautenticated

15 Error Estimates for te Finite Volume Element Metod 265 Proof. We find from [1, pp ], for ɛ small, wit C independent of ɛ, ϕ(z ) ϕ(z) sup C ϕ Lp(B), wit p = 2/ɛ, ϕ W 1 z,z B, z z z z 1 ɛ p (B). (4.15) We sall also apply te Sobolev inequality, wit explicit dependence on p, ϕ Lp(B) C p 1/2 ϕ H 1 (B), for p <, ϕ H 1 (B). (4.16) For ϕ H 1 (B) a proof was sketced in [12, Lemma 6.4]. For te general case of ϕ H 1 (B), we make a bounded extension of ϕ from H 1 (B) to H 1 ( B), wit B B, cf. [1, Capter IV] and apply (4.16) to H 1 ( B) to complete te proof. Employing (4.16) yields ϕ Lp(B) C p 1/2( ϕ H 1 (B) + ϕ H 2 (B)), ϕ H 2 (B). Combining tis wit (4.15), using p 1/2 = (2/ɛ) 1/2, completes te proof. Proof of Teorem 4.2. Te proof proceeds as tat of Teorem 4.1, starting wit (4.7) and noting tat te bound (4.8) for I remains valid. In order to bound II, we follow te steps above, but now, instead of (4.11), we sow M I ϕ(ζ ) C 3 l 1/2 ϕ H 2 (Π ζ ). (4.17) Using (4.17) as (4.11) in (4.12), we find [ψ, M I ϕ] C 2 l 1/2 ψ χ, ψ, χ S, (4.18) and ence II C 2 l 1/2 ψ. Togeter wit (4.8), tis completes te proof of (4.13). Te error estimate (4.14) now follows from Lemma 3.1 and Ẽ(t) Q P v Ct 1 Q P v C 2 l 1/2 t 1 v, for t >. It remains to sow (4.17). Let Π ζ be te symmetric patc associated wit Π ζ by te definition of almost symmetric. After a preliminary translation of Π ζ by O( 2 ), we may assume tat ζ = ζ. Furter, witout loss of generality, we may assume tat Π ζ Π ζ. In fact, if tis is not te case originally, it will be satisfied by srinking Π ζ by a suitable factor 1 c 2 wit c. Starting wit Π ζ we may now move te vertices one by one by O( 2 ) to obtain Π ζ in a finite number of steps, troug a sequence of intermediate patces Π ζ Π ζ. Applying (4.1) we will sow tat for eac of tese M Πζ I ϕ Cɛ 3 ɛ ϕ H 2 (Π ζ ), were C ɛ = Cɛ 1/2, ɛ >, (4.19) wic implies (4.17), by taking ɛ = l 1 and Π ζ = Π ζ. Since (4.19) olds for te symmetric patc Π ζ, by (4.11), it remains to sow tat if it olds for a given patc Π ζ ten it also olds for te next patc in te sequence. Assuming tus tat (4.19) olds for Π ζ, we consider te effect of moving one of its vertices, ζ 2, say, to ζ 2, wit ζ 2 ζ 2 = O( 2 ). Unautenticated

16 266 Panagiotis Catzipantelidis, Raytco Lazarov, Vidar Tomée Applying Lemma 4.2 to te function ϕ( ), wit B suitable, we obtain ϕ(z ) ϕ(z) sup C z,z Π ζ, z z z z 1 ɛ ɛ 1+ɛ( ) ϕ H 1 (Π ζ ) + ϕ H 2 (Π ζ ) C ɛ 1+ɛ ϕ H 2 (Π ζ ). (4.2) Moving only te vertex ζ 2 in Π ζ canges only te triangles τ 1 and τ 2 and tus te terms corresponding to j = 1, 2, 3 in (4.1). Letting τ 1 and τ 2 be te new triangles, te cange in te term wit j = 1 is ten bounded, since τ 1 τ 1 C 3, by (ω 1 ω 1 ) ( ϕ(ζ 1 ) ϕ(ζ ) ) C τ 1 τ 1 1 ɛ ϕ(ζ 1) ϕ(ζ ) ζ 1 ζ 1 ɛ C ɛ 3 ϕ H 2 (Π ζ ), and tus by te rigt-and side of (4.19). Te cange in te term wit j = 3 is bounded in te same way. For j = 2 te cange is bounded by te modulus of ω 2( ϕ(ζ 2 ) ϕ(ζ ) ) ω 2 ( ϕ(ζ2 ) ϕ(ζ ) ) = (ω 2 ω 2 ) ( ϕ(ζ 2 ) ϕ(ζ ) ) + ω 2( ϕ(ζ 2 ) ϕ(ζ 2 ) ). Te first term on te rigt is bounded as te terms wit j = 1, 3, and te second is bounded, using (4.2), since ζ 2 ζ 2 C 2, in te following way, ω 2 ( ϕ(ζ 2 ) ϕ(ζ 2 ) ) Cɛ 2 ζ 2 ζ 2 1 ɛ 1+ɛ ϕ H 2 (Π ζ ) C ɛ 3 ɛ ϕ H 2 (Π ζ ). Tis sows tat (4.19) remains valid after moving ζ 2, wic concludes te proof. More generally, we sall consider families of piecewise almost symmetric triangulations {T }, in wic Ω is partitioned into a fixed set of subdomains {Ω k } K k=1, and eac of tese is supplied wit an almost symmetric family {T (Ω k )} so tat T families may be constructed for any convex polygonal domain, cf. Figure 4, by successively refining an initial coarse mes, a procedure routinely used in computational practice. For suc meses we sow te following result. Teorem 4.3. If te family {T } is piecewise almost symmetric, ten Hence, for te solution of (2.4) wit v = P v, we ave = K k=1 T (Ω k ). Suc Q ψ C 3/2 ψ, ψ S. (4.21) ũ (t) u(t) C 3/2 t 1 v, for t >. (4.22) Proof. Following again te steps in te proof of Teorem 4.1, we note tat (4.8) still olds, and it remains to bound II. For eac internal vertex ζ of one of te T (Ω k ), te corresponding patc Π ζ is a O( 2 ) perturbation of a symmetric patc, and tus (4.17) olds. For ζ Z a vertex on te boundary of two of te T (Ω k ) we see tat by (4.1) M χ(ζ ) C 3 max x Π ζ χ(x) C 2 χ L2 (Π ζ ), Unautenticated

17 Error Estimates for te Finite Volume Element Metod 267 and by te use of approximation properties of te interpolation operator I we get M I ϕ(ζ ) C 2 I ϕ H 1 (Π ζ ) C 2( ϕ H 1 (Π ζ ) + ϕ H 2 (Π ζ )). (4.23) Using (4.17) and (4.23) as earlier (4.11) in (4.12), we conclude [ψ, M I ϕ] C 2 l 1/2 ψ ϕ 2 + C ψ ϕ H 1 (Ω S ), were Ω S is a strip of widt O() around te interface between te subdomains Ω k of Ω. Using now te inequality ϕ H 1 (Ω S ) C 1/2 ϕ H 2 (Ω) C 1/2 χ, we get [ψ, M I ϕ] C 3/2 ψ χ, ψ, χ S, (4.24) and ence II C 3/2 ψ. Togeter wit (4.8), tis completes te proof of (4.21). Te error estimate (4.22) now follows by Lemma 3.1 and Ẽ(t) Q P v Ct 1 Q P v C 3/2 t 1 v, for t >. We remark tat te operator M used ere, modulo a constant factor, is te same as te operator in [4]. Te arguments in te proofs of Teorems 4.2 and 4.3 terefore sow tat te following result olds for te lumped mass metod. Corollary 4.1. Assume tat {T } is almost or piecewise almost symmetric. Ten te nonsmoot data error estimates for te lumped mass metod, corresponding to (4.13) and (4.21), respectively, old. We finis tis section by remarking tat, in one space dimension, te full O( 2 ) L 2 -norm bound (1.17) for Q olds also for almost symmetric partitions, witout a logaritmic factor. Let Ω = (, 1) be partitioned by = x < x 1 < < x N +1 = 1. Denote now T = {τ i } N +1 i=1, wit τ i = [x i 1, x i ], and let S be te set of te continuous piecewise linear functions over T, vanising at x =, 1. We set i = x i x i 1 and = max i i. Te control volumes are V i = (x i i /2, x i + i+1 /2) and J ψ(x) = ψ(x i ) for x V i. We say tat T is almost symmetric if i+1 i C 2 for all i. Simple calculations sow, wit (χ, ψ) = 1 χψ dx and χ, ψ = (χ, J ψ), for χ, ψ S, ε (ψ, χ) = ψ, χ (ψ, χ) = 1 24 N i=1 ψ i ( i+1 (χ i+1 χ i ) i (χ i χ i 1 ) ), were w i = w(x i ) for a function w on Ω, and te one-dimensional version of (4.1) at x i becomes M I ϕ(x i ) = 1 24( i+1 (ϕ i+1 ϕ i ) + i (ϕ i 1 ϕ i ) ), i = 1,..., N. Te crucial step to prove (1.17) is ten to sow an analogue of (4.11), in tis case M I ϕ(x i ) C 5/2 ϕ H 2 (Π xi ), i = 1,..., N, wit Π xi = τ i τ i+1, (4.25) from wic (1.17) follows as earlier. Using te Taylor formula ϕ(x) = ϕ(x i ) + (x x i )ϕ (x i ) + x x i (x y) ϕ (y) dy, Unautenticated

18 268 Panagiotis Catzipantelidis, Raytco Lazarov, Vidar Tomée we find easily M I ϕ(x i ) = 1 24 (2 i+1 2 i )ϕ (x i ) + O ( 5/2 ϕ L2 (Π xi )), i = 1,..., N. By te almost symmetry, 2 i+1 2 i C 3 and by te Sobolev type inequality ϕ (x i ) C 1/2( ϕ L2 (Π xi ) + ϕ L2 (Π xi )) C 1/2 ϕ H 2 (Π xi ), for i = 1,..., N, we now conclude tat (4.25) olds. 5. Examples of Nonoptimal Nonsmoot Initial Data Estimates In tis section we present two examples were te necessary and sufficiency condition (3.6) for an optimal O( 2 ) nonsmoot data error estimate for t > is not satisfied. In te first example we construct a family of nonsymmetric meses {T } for wic te norm on te left-and side of (3.6) is bounded below by c, tus sowing tat te first order error bound of Teorem 3.5 is te best possible. In te second example we exibit a piecewise symmetric mes for wic tis norm is bounded below by c 3/2, implying tat te error estimate of Teorem 4.3 is best possible. In our first example we coose Ω = (, 1) (, 1) and introduce a quasi-uniform family of triangulations {T } of Ω as follows. Let N be a positive integer divisible by 4, = 4/(3N), x =, and set, for j = 1,..., N and m =, 1,..., M = 3 4 N, x j = x j 1 + { 1, 2 for j odd,, for j even, and y m = m. (5.1) We split te rectangle (x j, x j+1 ) (y m, y m+1 ) into two triangles by connecting te nodes (x j, y m ) and (x j+1, y m 1 ), see Figure 5. Tis defines a triangulation T tat is not symmetric at any vertex. Let now ζ = (x 2j, y m ), ζ Z, and let Π ζ be te corresponding nonsymmetric patc sown in Figure 5, wit vertices {ζ j } 6 j=1. Let τ j be te triangle in Π ζ wit vertices ζ, ζ j, ζ j+1, were ζ 7 = ζ 1. We ten ave τ j = 1 4 2, for j = 1, 2, 3, and τ j = 1 2 2, for j = 4, 5, 6. Tus, using (4.1), for ψ S, we obtain wit ψ j = ψ(ζ j ), M ψ(ζ ) = 1 6 ω j (ψ j ψ ) = 1 2 ( 3(ψ 1 + ψ 4 2ψ ) j=1 ) + 2(ψ 2 ψ ) + 2(ψ 3 ψ ) + 4(ψ 5 ψ ) + 4(ψ 6 ψ ). (5.2) Because ψ is piecewise constant over Π ζ, we easily see tat (5.2) implies M ψ(ζ ) C 2 ψ L2 (Π ζ ), ψ S. (5.3) For a smoot function ϕ we ave, by Taylor expansion, ϕ(ζ j ) ϕ(ζ ) = ϕ(ζ ) (ζ j ζ ) + O( 2 ), Unautenticated

19 Error Estimates for te Finite Volume Element Metod ζ 5 ζ 4 y m y m τ 4 τ 5 τ 3 ζ 6 τ 6 ζ ζ 3 τ 2 τ x 2j 1 x 2j x 2j+1 x 2(j+1) ζ 1 x 2j 2 ζ 2 Figure 5. Left: A nonsymmetric mes. Rigt: A nonsymmetric patc Π ζ, around ζ. were ζ j is considered as a vector wit components its Cartesian coordinates and te dot denotes te Euclidean inner product in R 2. Employing tis in (5.2), we find, after a simple calculation, M I ϕ(ζ ) = 3 18 ϕ(ζ ) (3, 1) + O( 4 ). (5.4) Let φ 1 (x, y) = 2 sin(πx) sin(πy) be te eigenfunction of, corresponding to te smallest eigenvalue λ 1 = 2π 2. We ten easily find tat φ 1 (1/4, 1/4) (3, 1) = 2π. Hence, tere exists a square P = [1/4 d, 1/4 + d] 2, wit < d < 1/4, suc tat φ 1 (z) (3, 1) 1, z P. (5.5) Letting now z Z P we ten ave tat M I φ 1 (z) c 3, c >, for small. We sall prove te following proposition. Proposition 5.1. Let T be defined by (5.1), P = {z = (x 2j, y m ) P} and consider te initial value problem (2.4) wit v = z P Φ z, were Φ z S is te nodal basis function of S at z. Ten we ave, for small, Ẽ(t) Q v c(t) v, wit c(t) >, for t >. Proof. Letting λ j and φ j be te eigenvalues and eigenfunctions of, and using Parseval s relation in S, equipped wit,, we ave Ẽ(t) N Q v 2 = e 2t λ j Q v, φ j 2 e 2t λ 1 Q v, φ 1 2. (5.6) j=1 Combining (2.3), (1.15) and (4.1), we find Q v, ψ = ( Q v, ψ) = ε (v, ψ) = [v, M ψ], ψ S. (5.7) Note now tat for z P, te corresponding patc Π z as te same form as te patc Π ζ considered above. Tus employing (5.3) for ζ = z, we get, for ψ S, [v, M ψ] [Φz, M ψ] = M ψ(z) C ψ, (5.8) z P z P Unautenticated

20 27 Panagiotis Catzipantelidis, Raytco Lazarov, Vidar Tomée 1 y m x J 1 x J x J+1 1 Figure 6. A piecewise symmetric mes. were in te last inequality we ave used te fact tat te number of points in P is O(N 2 ) = O( 2 ). We recall from [8] tat φ 1 φ 1 H 1 = O() and λ 1 λ 1, as, and, since obviously φ 1 I φ 1 H 1 = O(), (5.8) wit ψ = φ 1 I φ 1 gives [v, M ( φ 1 I φ 1 )] C ( φ 1 I φ 1 ) C 2. (5.9) For every z P, (5.5) olds, and tus, using (5.4) wit ϕ = φ 1 and ζ = z, we obtain, for small, since te number of vertices in P is bounded below by cn 2, [v, M I φ 1 ] = z P M I φ 1 (z) c 3 N 2 = c, wit c >. Combining tis wit (5.9), we obtain, for small, [v, M φ 1 ] [v, M I φ 1 ] [v, M ( φ 1 I φ 1 )] c C 2 c, wit c >. Since v = O(1), (5.6) and (5.7) now sow Ẽ(t) Q v e t λ 1 [v, M φ 1 ] c(t) v, for t >. Since and are equivalent norms, te proof is complete. It follows from Proposition 5.1 and Lemma 3.1 tat te igest order of convergence tat can old, uniformly for all v L 2, and for any family of triangulations {T }, is O(), i.e., Teorem 3.4 is best possible, in tis case. We now turn to our second example, in wic {T } is a piecewise symmetric family. Let again Ω = (, 1) (, 1) and consider a triangulation T of Ω, were te nodes (x j, y m ) are given as follows. Wit J a positive integer, let N = 7J, M = 4J and = 1/(4J), and set for j =,..., N and m =,..., M, { j, for j J, x j = and y m = m, (5.1) 1/4 + (j J)/2, for J < j N, Unautenticated

21 Error Estimates for te Finite Volume Element Metod 271 see Figure 6. Tis time we consider te set of vertices in P wit x = 1/4 and prove te following proposition. Proposition 5.2. Let T be defined by (5.1) and P = {z = (x J, y m ) P}. For te initial value problem (2.4), wit v = z P Φ z, were Φ z S is te nodal basis function of S at z, we ave, for small, Ẽ(t) Q v c(t) 3/2 v, wit c(t) >, for t >. Proof. Again, using (5.6) and (5.7), we ave, Ẽ(t) Q v 2 e 2t λ 1 [v, M φ 1 ] 2. (5.11) For z P, te corresponding patc Π z as te same form as te patc Π ζ considered above, see Figure 5 (rigt). Tus employing (5.3) for ζ = z and taking into account tat te number of vertices in P is O(N) we now obtain, for ψ S, [v, M ψ] [Φz, M ψ] = M ψ(z) C 3/2 ψ. z P z P Similarly to (5.9) tis now sows and, again using (5.4), for small, [v, M I φ 1 ] = z P Combined wit (5.12) tis gives, for small, [v, M ( φ 1 I φ 1 )] C 5/2, (5.12) M I φ 1 (z) c 3 J = c 2, wit c >. [v, M φ 1 ] c 2 C 5/2 c 2, wit c >. (5.13) Since v = O( 1/2 ) we obtain from (5.11) and (5.13) Ẽ(t) Q v c(t) 2 c(t) 3/2 v, for t >. It follows from Proposition 5.2 and Lemma 3.1 tat te igest order of convergence tat can old, uniformly for all v L 2, and for all piecewise symmetric families {T }, is O( 3/2 ), i.e., Teorem 4.3 is best possible in tis regard. Remark 5.1. Since M is proportional to te operator used in [4], te arguments in tis section also apply to te lumped mass metod. In particular, te analogue of Proposition 5.1 ten sows tat te first order nonsmoot data estimate for t > of [4, Teorem 4.3] is best possible for general triangulations {T }. Furter, te O( 3/2 ) estimate stated in Corollary 4.1 is best possible for piecewise almost symmetric triangulations. Our examples ere may be tougt of as generalizations to two space dimensions of te one-dimensional counter-examples in [4, Section 7]. Unautenticated

22 272 Panagiotis Catzipantelidis, Raytco Lazarov, Vidar Tomée 6. Some Fully Discrete Scemes In tis section we discuss briefly te generalization of our above results for te spatially semidiscrete finite volume metod to some basic fully discrete scemes, namely te backward Euler and Crank Nicolson metods. Wit k >, t n = n k, n =, 1,..., te backward Euler finite volume metod approximates u(t n ) by Ũ n S for n suc tat, wit Ũ n = (Ũ n Ũ n 1 )/k, Ũ n, χ + ( Ũ n, χ) =, χ S, for n 1, wit Ũ = v, or, Ũ n Ũ n =, for n 1, wit Ũ = v. (6.1) Introducing te discrete solution operator Ẽk = (I k ) 1 we may write Ũ n = Ẽ k Ũ n 1 = Ẽn kũ, n 1. Using eigenfunction expansion and Parseval s relation, we obtain, analogously to [12, Capter 7], te stability property p Ẽ n kχ C p χ, χ S, for p =, 1. (6.2) Te estimates tat follow and teir proofs are analogous to tose for te lumped mass metod derived in [4], since te operators Ẽ(t), and Q, defined in Section 2, ave properties analogous to tose of te corresponding operators for te lumped mass metod. For simplicity we will only sketc te proof of Teorem 6.1. We sall use te following abstract lemma sown in [4], in te case H = S, normed by, and wit A =. Lemma 6.1. Let A be a linear, selfadjoint, positive definite operator in a Hilbert space H, wit compact inverse, let u = u(t) be te solution of and let U = {U n } n= be defined by u + Au =, for t >, wit u() = v, U n + AU n =, for n 1, wit U = v. Ten, for p =, 1, 1 q 3, wit p + q, we ave A p/2 (U n u(t n )) Ckt (1 q/2) n A (p+q)/2 v, for n 1. Te error estimates of te following teorem for (6.1) are of optimal order under te same assumptions as in Section 3. Teorem 6.1. Let u and Ũ be te solutions of (1.1) and (6.1). Ten, for n 1, C( 2 + k) v 2, if v v C 2 v 2, Ũ n u(t n ) C( 2 + k)t 1/2 n v 1, if v = P v and P v C v 1, C( 2 + k)t 1 n v, if v = P v and (1.17) olds. Unautenticated

23 Error Estimates for te Finite Volume Element Metod 273 Proof. Analogously to te proof of [4, Teorem 8.1], we split te error as Ũ n u(t n ) = (Ũ n ũ (t n )) + (ũ (t n ) u(t n )) = β n + η n. By Teorems , η n is bounded as required. In order to bound β n = (Ẽn k Ẽ(t n ))v in te smoot data case, it suffices, using te stability estimates (6.2) and Lemma 2.1, to consider v = R v. We obtain by Lemma 6.1, wit A = A =, and q = 2, 1,, β n = Ũ n ũ (t n ) Ckt (1 q/2) were for q = 2, te last inequality follows from n A q/2 v Cktn (1 q/2) v q, A R v 2 = ( R v, A R v) = ( v, A R v) = ( v, A R v), for q = 1 from A 1/2 P v = P v C v 1 and for q = from P v C v. Also for te lumped mass metod te analogous result in te mildly nonsmoot data case v Ḣ1 olds, and sould replace te result for q = 1 in [4, Teorem 8.1], cf. te remark after Teorem 3.2. Recall tat Q satisfies (1.17) if {T } is symmetric. For almost symmetric or piecewise almost symmetric {T } we obtain correspondingly te following nonsmoot initial data error estimates employing (4.14) and (4.22). Teorem 6.2. Let u and Ũ be te solutions of (1.1) and (6.1), wit v = P v. Ten, for n 1, { Ũ n C( 2 l 1/2 + k)t 1 n v, if {T } is almost symmetric, u(t n ) C( 3/2 + k)t 1 n v, if {T } is piecewise almost symmetric. For te gradient of te error we may prove as in [4, Teorem 8.2], te following smoot and nonsmoot data error estimates, witout additional assumptions on T. For smoot initial data we assumed in [4] tat v = R v, but te more general coices of v are permitted by te stability estimates (6.2) and Lemma 2.1. Teorem 6.3. Let u and Ũ be te solutions of (1.1) and (6.1). Ten, for n 1, { (Ũ n C( + k) v 3, if (v v) C v 2, u(t n )) C( t 1 n + k t 3/2 n ) v, if v = P v. We now turn to te Crank Nicolson metod, defined by Ũ n Ũ n 1 2 =, for n 1, wit U = v, Ũ n 1 2 = 1 2 (Ũ n + Ũ n 1 ). (6.3) Denoting again te discrete solution operator by Ẽk = (I+ 1 2 k )(I 1 2 k ) 1 we may write Ũ n = ẼkŨ n 1 = Ẽn kũ, n 1. Using eigenfunction expansion and Parseval s relation, we find tat (6.2) also olds for tis metod. Te Crank Nicolson metod does not ave as advantageous smooting properties as te backward Euler metod, wic is reflected in te fact tat te following analogue of Lemma 6.1, sown in [4, Lemma 8.2], does not allow q =. Unautenticated

24 274 Panagiotis Catzipantelidis, Raytco Lazarov, Vidar Tomée Lemma 6.2. Let A and u(t) be as in Lemma 6.1 and let U n satisfy Ten U n + AU n 1 2 =, for n 1, wit U = v. A p/2 (U n u(t n )) Ck 2 t (2 q) n A p/2+q v, for n 1, p =, 1, q = 1, 2. Tis time optimal order estimates for te error in L 2 and in H 1 old uniformly down to t =, if v Ḣ4 and v Ḣ5, respectively. Te proofs are analogous to tose of [4, Teorems 8.3 and 8.4], were we assumed v = R v. Again te stability estimates (6.2) and Lemma 2.1 permit te more general coices for v. Teorem 6.4. Let u and ave, for n 1, Ũ be te solutions of (1.1) and (6.3). Ten, wit q = 1, 2, we Ũ n u(t n ) C( 2 + k 2 t (2 q) n ) v 2q, if v v C 2 v 2, (Ũ n u(t n )) C( + k 2 t (2 q) n ) v 2q+1, if (v v) C v 2. For optimal order convergence for initial data only in L 2, one may modify te Crank Nicolson sceme by taking te first two steps by te backward Euler metod, wic as a smooting effect. We may sow ten te following result, analogously to tat of [4, Teorem 8.5], wit te obvious modifications for almost symmetric and piecewise almost symmetric families {T }. Teorem 6.5. Let u be te solution of (1.1) and Ũ n tat of (6.1), for n = 1, 2, and of (6.3), for n 3, wit v = P v and assume (1.17) olds. Ten we ave Ũ n u(t n ) C( 2 t 1 n + k 2 t 2 n ) v, for n Problems wit More General Elliptic Operators Tis final section is devoted to te extension of our earlier results to te more general problem (1.19), and we recall tat we sall consider te finite volume metod (1.26) were te bilinear form ã (, ) is defined by (1.25). Our error analysis is again based on estimates for te standard Galerkin finite element metod, in tis case defined by (1.2) and (1.21). It is well known tat for tis metod te stability and smooting estimates (2.2) old as do te error estimates (1.5) (1.7), were te norms q are defined analogously to te norms (1.2), using te eigenvalues and eigenfunctions of A. We introduce te discrete elliptic operator à : S S by Ãψ, χ = ã (ψ, J χ), χ, ψ S, (7.1) wic is symmetric and positive definite wit respect to te inner product, by (1.24), since ( βψ, J χ) is symmetric, positive semidefinite on S. Tis follows from te fact tat τ χj ψ dx is symmetric by (4.6) and β is constant and non-negative in eac τ of T. We may ten rewrite (1.26) as ũ,t + Ãũ =, for t, wit ũ () = v, (7.2) Unautenticated

25 Error Estimates for te Finite Volume Element Metod 275 and te solution is given by ũ (t) = Ẽ(t)v, were Ẽ(t) = e à t is defined as in (2.5), wit { λ j } and { φ j } te eigenvalues and eigenfunctions of Ã, ortonormal wit respect to,. Note tat a sligtly different finite volume element metod for (1.19) as been considered in [9]. Tis metod differs in te discretization of te lower order term, using te bilinear ā (, ) defined by ā (ψ, J χ) = ( α ψ, χ) + (βj ψ, J χ), ψ, χ S. For tis metod analogous results to Teorems old. Following our error analysis in te previous sections we introduce δ = ũ u and split te error into ũ u = δ + (u u), were u u and (u u) are estimated by te analogues of (1.5) (1.7). It terefore suffices to derive estimates for δ, wic satisfies, for t, δ,t, χ + ã (δ, J χ) = ε (u,t, χ) ε (u, χ), χ S, wit δ() =, (7.3) were ε (, ) is given by (1.16) and ε (, ) is defined by ε (ψ, χ) = ã (ψ, J χ) a(ψ, χ), ψ, χ S. (7.4) Now let Q : S S and Q : S S be te quadrature error operators given by ã (Q ψ, J χ) = ε (ψ, χ) and ã ( Q ψ, J χ) = ε (ψ, χ), ψ, χ S. (7.5) Using (7.1), te equation (7.3) for δ can ten be written in operator form as δ t + Ãδ = ÃQ u,t à Q u, for t, wit δ() =. Tis problem is similar to (3.1), except tat te operator is replaced by à and tat on te rigt-and side we ave an additional term resulting from te approximation of te bilinear form a(, ). By Duamel s principle we ave δ(t) = t Ẽ (t s)ãq u,t (s) ds t Ẽ (t s)ã Q u (s) ds =: δ(t) + δ(t), for t. (7.6) To estimate δ it terefore suffices to bound δ and δ. For tis we need some auxiliary results, wic are discussed below. Lemma 7.1. Let α, β C 2. For te error functional ε, defined by (7.4), we ave ε (ψ, χ) C p+q q ψ p χ, ψ, χ S, wit p, q =, 1. Proof. In view of (7.4), we may write ε (ψ, χ) = (( α α) ψ, χ) + ( βψ, J χ) (βψ, χ). We ten split ε (ψ, χ) as a sum of integrals over τ T. Since α = α(z τ ), we see tat (f f(z τ τ))dx = for linear functions f, and ence (f f(z τ ))dx C 2 τ τ f C 2, for f C 2, (7.7) τ Unautenticated