Error estimates for a finite volume element method for parabolic equations in convex polygonal domains

Transcription

1 Error estimates for a finite volume element method for parabolic equations in convex polygonal domains P Chatzipantelidis, R D Lazarov Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA V Thomée Department of Mathematics, Chalmers University of Technology, S Göteborg, Sweden Received XXXXXXXXX We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method Because the domain is polygonal special attention has to be paid to the limited regularity of the exact solution We give sufficient conditions in terms of data which yield optimal order error estimates in L 2 and H 1 The convergence rate in the L norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners We also briefly consider the lumped mass modification and the backward Euler fully discrete method c 24 John Wiley & Sons, Inc Keywords: finite volume element method, parabolic equation, error estimates, elliptic projection I INTRODUCTION We consider the model initial boundary value problem u t Lu = f, in Ω, t, u =, on Ω, t, (11 u( = u, in Ω, where Ω is a bounded, convex, polygonal domain in R 2 and Lu div(a u, with A = (a ij 2 i,j=1 a real valued smooth matrix function, uniformly positive definite in Ω and f : Ω [, R We shall study spatially semidiscrete approximations of (11 by the finite volume element method, which for brevity we will refer to as the finite volume method below Numerical Methods for Partial Differential Equations X, No X, 1 21 (24 c 24 John Wiley & Sons, Inc CCC XXXX-XXXX/4/XXXXXXXXX

2 2 P CHATZIPANTELIDIS, R D LAZAROV AND V THOMÉE V z z K z K K z z FIG 1 Left: A union of triangles that have a common vertex z; the dotted line shows the boundary of the corresponding control volume V z Right: A triangle K partitioned into the three subregions K z The approximate solution will be sought in the piecewise linear finite element space X h X h (Ω = {χ C(Ω : χ K linear, K T h ; χ Ω = }, where {T h } <h<1 is a family of regular triangulations of Ω, with h denoting the maximum diameter of the triangles of T h As a model for our analysis we shall first consider the corresponding semidiscrete Galerkin finite element method, which is to find u h (t X h such that, with (, the inner product in L 2 (Ω, (u h,t, χ a(u h, χ = (f, χ, χ X h, t, u h ( = u h, (12 where u h X h is a given approximation of u and the bilinear form a(, is defined by a(v, w = Ω A v w dx, for v, w H 1 (Ω The finite volume method that we consider is instead designed to inherit a local conservation property associated with the differential equation Namely, integrating (11 over any region V Ω and using Green s formula, we obtain u t dx (A u n ds = f dx, t, (13 V V V where n denotes the unit exterior normal to V The semidiscrete finite volume problem will satisfy a relation similar to (13 for V in a finite collection of subregions of Ω called control volumes, the number of which will be equal to the dimension of the finite element space X h These control volumes are constructed in the following way Let z K be the barycenter of K T h We connect z K with line segments to the midpoints of the edges of K, thus partitioning K into three quadrilaterals K z, z Z h (K, where Z h (K are the vertices of K Then with each vertex z Z h = K Th Z h (K we associate a control volume V z, which consists of the union of the subregions K z, sharing the vertex z (see Figure 1 We denote the set of interior vertices of Z h by Z h

3 FVEM ERROR ESTIMATES FOR PARABOLIC PDE S IN POLYGONS 3 The semidiscrete finite volume method is then to find u h (t X h for t such that u h,t dx V z (A u h n ds = V z f dx, V z z Zh, t, (14 u h ( = u h, with u h X h a given approximation of u This version of the finite volume method is also referred to as a vertex centered finite volume method Similar discretization techniques have been analyzed for various linear and nonlinear evolution problems, cf, eg, [1, 2, 3] The finite volume problem (14 can be rewritten in a variational form similar to the finite element problem (12 For this purpose we introduce the finite dimensional space Y h = {η L 2 (Ω : η Vz = constant, z Z h; η Vz =, z Ω} For an arbitrary η Y h, we multiply the integral relation in (14 by η(z and sum over all z Z h to obtain the Petrov Galerkin formulation, to find u h(t X h for t such that (u h,t, η a h (u h, η = (f, η, η Y h, t, u h ( = u h, (15 where the bilinear form a h (, : X h Y h R is defined by a h (v, η = η(z (A v n ds, v X h, η Y h (16 V z z Z h Obviously, a h (v, η may be defined by (16 also for v H 2 (Ω, and using Green s formula we easily see that a h (v, η = (Lv, η, for v H 2 (Ω, η Y h (17 In the same way as for the finite element method, the finite volume method (15 may be written as a system of ordinary differential equations In fact, let {Φ z } z Z h be the standard pyramid basis of X h, with Φ z (w = 1 if w = z and Φ z (w = if w z, w Zh, and let {Ψ z} z Z h be the associated basis of Y h consisting of the characteristic functions of the control volumes V z Writing u h (t = z Z α h z (tφ z, (15 then takes the form Mα (t Sα(t = f(t, t, with α( = α, (18 where M = (m zw and S = (s zw are the associated mass and stiffness matrices, respectively, with m zw = (Φ z, Ψ w and s zw = a h (Φ z, Ψ w, and where α(t, α, and f(t have elements α z (t, (u, Ψ z, and (f(t, Ψ z It is easy to see that M is symmetric and that both M and S are positive definite, see Section 3 below The above approach was perhaps first formulated in the Petrov-Galerkin framework, employing two different meshes to define the solution space and the test space, for elliptic equations of second order, by Bank and Rose [4], Cai [5], and Süli [6] This approach allows us to analyze the error by using some of the tools developed in the finite element method theory Other finite volume methods have also been considered in the literature, such as what are referred to as finite volume difference methods, using cell centered grids and

4 4 P CHATZIPANTELIDIS, R D LAZAROV AND V THOMÉE approximating derivatives by finite differences in the balance equation, see eg, the survey papers [2, 3] These are not covered by our analysis In the existing literature the error estimates for the finite volume method are usually derived for solutions that are sufficiently smooth, cf, eg, [1, 2, 3, 7, 8] However, in many applications the solutions are not smooth due to, eg, nonsmooth diffusion coefficient (this is often called an interface problem, nonsmooth right-hand side, incompatible initial data, and domains with nonsmooth boundaries, eg, polygonal domains Recently, Ewing et al [9], Droniou and Galouët [1], and two of the present authors [11, 12] considered finite volume schemes for elliptic problems with nonsmooth right-hand side, namely f H l (Ω with l > In this paper we present an error analysis of the finite volume method for parabolic problems in convex polygonal domains Before we start our technical description of this work we introduce some notation We shall denote by L p (V the p integrable real valued functions over V R 2, (, V the inner product in L 2 (V, and W s p (V the norm in the Sobolev space W p s (V, s If V = Ω we suppress the index V, and if p = 2 we write H s = W2 s and = L2 Finally, we denote by L q (, T ; Wp s, 1 p, q, s, the space of all measurable functions v(t : [, T ] Wp s such that u(t W s p L q (, T (see, eg [13, p 285] As a guide to our analysis we begin with some comments about error estimates for the finite element method (12; details will be given in Section II below It is well known, cf, eg, [14, Chapter 1], that u h (t u(t u h u Ch 2( u H 2 u t H 2 dτ, t, (19 where we assume that u is smooth enough for the right-hand side to be finite This error bound is shown in [14] for domains with smooth boundary, but the proof is valid also for a polygonal domain If u h is chosen so that u h u Ch 2 u H 2, the first term on the right may be bounded by the second In the sequel we shall often make such particular choices, to avoid having to account for this term In the proof of (19 one splits the error as u h u = (u h R h u (R h u u where R h denotes the standard Ritz projection R h : H 1 X h defined by a(r h u, χ = a(u, χ, χ X h (11 It is well known that R h has the approximation property R h v v H j Ch l j v H l, v H l H 1, j =, 1, l = 1, 2 (111 As mentioned above, error bounds in this area are normally expressed in terms of norms of the exact solution of the problem, such as is the case in (19 In the case of a domain with smooth boundary it is well known that the regularity required of the exact solution can be attained by assuming enough regularity of the data Thus there is no great need to express the error bounds in terms of data, which would be an unnecessary complication in the analysis In the case of a polygonal domain the situation is different The corners of the domain then give rise to singularities in the solution that are present regardless of the smoothness of the data It then becomes an essential part of the analysis to show that the norms appearing in a proposed optimal order error estimate can be guaranteed to be finite under the appropriate smoothness and possible additional assumptions on data Such an additional assumption could be a compatibility relation between f and u on Ω for

5 FVEM ERROR ESTIMATES FOR PARABOLIC PDE S IN POLYGONS 5 t = We have therefore chosen here to express our error estimates in terms of norms of data, sometimes supplemented with compatibility conditions As an example, in order for (19 to show a O(h 2 error estimate we need to show that u t L 1 (, T ; H 2 for some T > We recall that since the domain is convex, we have the elliptic regularity estimate (see, eg, [15] u H 2 C Lu (112 Using this together with the differential equation (11 we shall show u t H 2 dτ C T ε 1 ( g H ε f t H ε dτ, t T, (113 for ε (, 1/2, where g = u t ( = f( Lu Thus the L 2 error in (19 is of order O(h 2 for t T if u H 2 and the right hand side of (113 is finite We remark that precise regularity estimates have been derived for elliptic problems in nonconvex domains, which gives hope of finding precise error estimates in our problem of the appropriate lower order than O(h 2 In the elliptic case such results were obtained for finite volume methods in [11, 12] In Section III we turn to the finite volume method (15 It was shown in Ewing, Lazarov, and Lin [8] and Li, Chen, and Wu [16] that, for p > 1 and u h appropriately chosen, u h (t u(t Ch 2( u W 3 p u t W 3 p dτ, t (114 This time the proof uses a splitting of the error based on the elliptic projection operator R h : H 2 H 1 X h defined by a h ( R h v, η = a h (v, η, η Y h (115 In Chou and Li [7] and Li, Chen, and Wu [16] it was shown that R h v v Ch 2 v W 3 p, for p > 1 (116 The right hand side here is bounded for p close to 1, which is expressed by the elliptic regularity estimate, cf [15, Chapter 5], v W 3 p C p Lv W 1 p, 1 < p < p 2, v W 3 p H 1, (117 where p is defined in the following way: Let S be a vertex of Ω, and denote the corresponding interior angle of Ω by ω(s Let A and T be matrices such that A = (a ij (S 2 i,j=1 and T AT T = I, and let ω A (S be the angle at the vertex T S of the transformed domain T Ω = {T x : x Ω} Define ω = max S ω A(S, β = π/ω, and p = 2/(3 β (118 We remark that ω < π, and that ω A (S = ω(s if L = Note that for a general convex polygonal domain (117 does not hold for p = 2 Also here A should be a smoother function than in the elliptic regularity estimate (112 In the sequel, we shall not give the precise regularity needed for A, but we will consider it to be sufficiently smooth To obtain a O(h 2 error bound for the finite volume method (15, we shall

6 6 P CHATZIPANTELIDIS, R D LAZAROV AND V THOMÉE demonstrate this time that, with g 1 = u tt ( = f t ( Lg, we have for p < p, u t W 3 p dτ C ( g 1 Lp f tt Lp dτ, t T (119 Section IV is devoted to an alternative way to obtain a O(h 2 error bound for the finite volume method (15, under slightly different regularity assumptions, which is to base the analysis on the standard Ritz projection R h defined in (11 rather than R h from (115 In this case we are able to show u h (t u(t Ch 2( u H 2 ( ( u t 2 H 2 f 2 H 1dτ 1/2, t T After establishing the a priori estimate u t 2 H 2 dτ C( g 2 H 1 f t 2 dτ, t T, under the compatibility condition g = on Ω, we obtain the desired O(h 2 error bound, under slightly different conditions than those in (119 In Section V we apply the technique of Section IV to derive error bounds in H 1 and L norms In the latter case we show a bound of the form u h (t u(t L C T,γ (u, f h γ, t T, for any γ < β (12 We remark that a O(h 2 log 1 h uniform bound for the error was shown in [7] under the assumption that u L (, T ; H 3 for T >, in the case of a smooth boundary However, as indicated above, this is not a realistic assumption for a polygonal domain However, the significant loss of convergence rate expressed in (12 only takes place near the corners of Ω, and away from these we are able to show a O(h 2 log 1 h maximum norm error bound In Section VI we consider a lumped mass finite volume element method In this case the method loses the property of being locally conservative For a constant coefficient matrix A, it reduces to a linear system with the same left-hand side as the corresponding lumped mass finite element method We analyze the lumped mass method using the elliptic projection R h and obtain the same error bounds as for (15 Finally, in Section VII, we show that our approach also applies to fully discrete schemes As an example we analyze the backward Euler finite volume element method II THE FINITE ELEMENT METHOD In this section, as a guide to the proofs of our subsequent error bounds for the finite volume method, we shall show the following result for the standard finite element method Theorem 21 Let u h and u be the solutions of (12 and (11, respectively, and assume that u = on Ω and ε (, 1/2 Then, if u h = R hu, u h (t u(t C T,ε (u, fh 2, t T, (21

7 FVEM ERROR ESTIMATES FOR PARABOLIC PDE S IN POLYGONS 7 where C T,ε (u, f = C T ( u H 2 ε 1( g H ε f t H ε dτ, and g = f( Lu Proof In a standard way we split the error u h u using the Ritz projection R h u, defined in (11, as In view of (111 we have u h u = (u h R h u (R h u u = ϑ ϱ ϱ(t Ch 2 u(t H 2 Ch 2 ( u H 2 In order to bound ϑ, we note that by our definitions (ϑ t, χ a(ϑ, χ = (ϱ t, χ, χ X h u t H 2 dτ, t (22 Choosing χ = ϑ and using the positivity of a(ϑ, ϑ, (111, and the fact that ϑ( =, we easily find ϑ(t C Thus, together with (22 we have u h (t u(t Ch 2 ( u H 2 ϱ t dτ Ch 2 u t H 2 dτ, t u t H 2 dτ, t The theorem therefore follows from the inequality (113 which we now proceed to show For this we use the elliptic regularity estimate (112 and differentiate (11 to obtain After integration it now remains to show u t H 2 C Lu t C( u tt f t (23 u tt dτ C T ε 1 ( g H ε Recall that, by Duhamel s principle, we have u(t = E(tu E(t sf(s ds = E(tu f t H ε dτ, t T (24 E(sf(t s ds, (25 where E(t denotes the solution operator of the homogeneous case (f = of (11 It is well-known that E(t is the analytic semigroup e Lt in L 2 generated by L, and that E(tv C v and E (tv Ct 1 v, (26 where the inequalities easily follow, using Parseval s relation, by writing E(tv = e λjt (v, φ j φ j, j=1

8 8 P CHATZIPANTELIDIS, R D LAZAROV AND V THOMÉE with {λ j } and {φ j } the eigenvalues and eigenfunctions of L Differentiating (25 twice in time we find and u t (t = E(tg u tt (t = E (tg f t (t Here we have that for ε (, 1/2 For this we first note that E(t sf t (s ds, (27 E (t sf t (s ds (28 E (tv Ct 1ε/2 v H ε, t > (29 E (tv C E(tLv C v H 2, v H 2 H 1 (21 By interpolation between this estimate and the second estimate in (26, (29 follows upon noting that [L 2, H 2 H 1 ] ε/2,2 = H ε, cf, eg, [15, Corollary 1445] Here [X, Y ] θ,q, θ 1, 1 q, denote the Banach spaces intermediate between X and Y defined by the K functional, which are used in interpolation theory, cf, eg, [17, Chapter 5] Finally, applying (29 in (28 we find ( u tt dτ C τ 1ε/2 g H ε dτ τ Ct ε/2 ε 1 ( g H ε f t dτ (τ s 1ε/2 f t (s H ε ds dτ f t H ε dτ, t (211 which shows (24 and hence (113 This completes the proof Note that since (29 has been obtained by interpolation between L 2 and H 2 H 1, the argument requires that v = on Ω, if ε (1/2, 2, but not for ε (, 1/2 Thus, in addition to reducing the regularity requirements on g and f t, compared to direct application of (21, using (29 makes it possible to avoid imposing unnatural boundary conditions on these functions III THE FINITE VOLUME METHOD In this section we begin, for completeness, by recalling some known preliminary material concerning the finite volume method, cf eg, [2, 4, 7, 16, 18, 19], and then proceed with an error bound The proof of the latter follows the lines of Theorem 21 for the finite element method, but uses the elliptic projection associated with the finite volume bilinear form For the analysis we introduce the interpolation operator I h : C(Ω Y h, defined by I h v = v(zψ z, z Z h

9 FVEM ERROR ESTIMATES FOR PARABOLIC PDE S IN POLYGONS 9 where as before Ψ z is the characteristic function of V z We note that I h : X h Y h is a bijection and bounded with respect to the L 2 norm, and that Ψ z = I h Φ z for z Z h Further, the bilinear form (χ, I h ψ is symmetric positive definite, thus an inner product on X h, and the corresponding discrete norm is equivalent to the L 2 norm, uniformly in h, ie, with c >, c χ χ C χ, χ X h, where χ (χ, I h χ 1/2 (31 In fact, for z and w neighboring vertices in Zh, we have, with W z = supp Φ z, m zw = (Φ z, Ψ w = Φ z dx = 7 W z V w 36 W z V w = 7 18 W z W w which is symmetric in z and w From this we find m zw 7 18 V z, (32 w z with equality when z is not a neighbor of a vertex of Ω Since m zz = Φ z dx = 11 V z 18 V z, (33 we conclude that the mass matrix M, cf (18, is diagonally dominant which easily shows (31 The bilinear form a h (, of (16 may equivalently be written as a h (χ, η = K ( (Lχ, ηk (A χ n, η K, χ Xh, η Y h, (34 where the quantity A χ in (, K is the trace on K of A χ on K Indeed, by integration by parts, we obtain, for z Zh and K T h, Lχ dx A χ n ds = A χ n ds K z K z K K z V z and (34 hence follows by multiplication by η(z and by summation first over the triangles that have z as a vertex and then over the vertices z Zh Also, we easily see that, for any side e of a triangle in T h, I h χ ds = χ ds, χ X h e Therefore, in the case of a constant coefficient matrix A we have e a h (χ, I h ψ = a(χ, ψ, χ, ψ X h, (35 since Lχ = For a smooth variable coefficient matrix A, one easily finds, cf, eg, [18, Lemma 52], [19, Lemma 2], a(χ, ψ a h (χ, I h ψ Ch χ H 1 ψ H 1, χ, ψ X h It follows that there exist positive constants c and h such that a h (χ, I h χ c χ 2 H 1, χ X h, h < h (36

10 1 P CHATZIPANTELIDIS, R D LAZAROV AND V THOMÉE In particular, this shows that the stiffness matrix S, cf (18, is positive definite In view of the above, the Petrov-Galerkin equation in (15 may also be written in Galerkin form, (u h,t, I h χ a h (u h, I h χ = (f, I h χ, χ X h, t We now turn to the error estimate for the finite volume method Theorem 31 Let u h and u be the solutions of (15 and (11, respectively, and assume that u = g = on Ω and 1 < p < p, with p as in (118 Then, if u h = R h u, where R h is the elliptic projection defined by (115, where u h (t u(t C T,p (u, fh 2, t T, (37 C T,p (u, f = C T,p ( u W 3 p g 1 Lp ( f t W 1 p f tt Lp dτ Proof For the convenience of the reader we briefly show the error estimate (114 from [8, 16] This time we write u h u = (u h R h u ( R h u u ϑ ϱ In view of (116 we have ϱ(t Ch 2 u(t W 3 p Ch 2 ( u W 3 p In order to bound ϑ we note that ( ϑ t, η a h ( ϑ, η = ( ϱ t, η, η Y h u t W 3 p dτ, t (38 Choosing η = I h ϑ and using the positivity of ah ( ϑ, I h ϑ, (116, and the fact that ϑ( =, we find easily ϑ(t C Together with (38 and (117, this shows at once u h (t u(t Ch 2( u W 3 p ϱ t dτ Ch 2 u t W 3 dτ, t p ( u tt W 1 p f t W 1 p dτ, t To prove the theorem it now remains to demonstrate the regularity estimate u tt W 1 p dτ C ( g 1 Lp f tt Lp dτ, t T If g W 2 p H 1, g 1 L p, and f tt L 1 (, T ; L p, differentiation in time of (27 shows u tt (t = E(tg 1 E(t sf tt (s ds, t, (39 where we have used f t ( Lg = f t ( Lu t ( = u tt ( = g 1 One can show that E(t is an analytic semigroup not only in L 2 but also in L p, for 1 < p <, so that E(tv Lp C v Lp and E (tv Lp C p t 1 v Lp, 1 < p < (31

11 FVEM ERROR ESTIMATES FOR PARABOLIC PDE S IN POLYGONS 11 This is proved in [2, Theorem 36, Chapter 7] for a domain with smooth boundary, but the arguments are easily seen to be valid also for a polygonal domain In view of (31 and (117, we have for p < p, cf (117, E(tv W 2 p C LE(tv Lp C E (tv Lp Ct 1 v Lp By interpolation between this and the first estimate in (31, and noting that [L p, W 2 p ] 1/2,1 W 1 p, cf [17, Chapter 5], [21, Chapter 12], we get Therefore E(tv W 1 p C E(tv [Lp,W 2 p ] 1/2,1 Ct 1/2 v Lp (311 u tt W 1 p dτ C ( τ 1/2 g 1 Lp dτ Ct 1/2 ( g 1 Lp τ f tt Lp dτ, t > (τ s 1/2 f tt (s Lp ds dτ We remark that instead the factor t 1/2 in the right hand side of (311, one could have an estimate with t 1ε as in (29 in the proof of Theorem 21 However, since the estimate would then involve negative norms, we will refrain from elaborating on this IV ALTERNATIVE ANALYSIS OF THE FINITE VOLUME METHOD In this section we give an alternative analysis of the finite volume method (15, in which the elliptic projection employed is the standard Ritz projection rather than the one based on the bilinear form a h (, This time we shall show the following error estimate Theorem 41 Let u h and u be the solutions of (15 and (11, respectively, and assume u = g = on Ω and ε (, 1/2 Then, if u h = R hu, with R h defined by (11, we have for t T, where u h (t u(t C T,ε (u, fh 2, (41 C T,ε (u ( t, f = C T u H 2 g H 1 ε 1 f t H ε dτ ( ( f t 2 f 2 H 1 dτ 1/2 Proof This time we split the error as u h u = (u h R h u (R h u u = ϑ ϱ We could use (22 and (113 to estimate ϱ as before, but since we will need more regularity from data in bounding θ than in the finite element method, we modify (211 by using ε = 1 in the term in g and obtain ϱ(t C T h 2( u H 2 g H 1 ε 1 f t H ε dτ, t T (42 We now turn to the estimate for ϑ Using (17, see that the error u h u satisfies ((u h u t, η a h (u h u, η = for all η Y h and thus we have for ϑ (ϑ t, η a h (ϑ, η = (ϱ t, η a h (ϱ, η, η Y h (43

12 12 P CHATZIPANTELIDIS, R D LAZAROV AND V THOMÉE For the analysis we introduce the error functionals We note that ε h (f, χ = (f, χ (f, I h χ, f L 2, χ X h, (44 ε a (χ, ψ = a(χ, ψ a h (χ, I h ψ, χ, ψ X h a h (R h u, I h ϑ = a(r h u, ϑ ε a (R h u, ϑ = a(u, ϑ ε a (R h u, ϑ = (Lu, ϑ ε a (R h u, ϑ, and hence, since a h (u, I h ϑ = (Lu, I h ϑ by (17, a h (ϱ, I h ϑ = ε h (Lu, ϑ ε a (R h u, ϑ (45 Choosing η = I h ϑ in (43 we therefore have, since Lu = f u t, (ϑ t, I h ϑ a h (ϑ, I h ϑ = (ϱ t, I h ϑ ε h (u t f, ϑ ε a (R h u, ϑ (46 The following bounds for ε h (, and ε a (, are shown in [18, Lemmas 51 and 52] Lemma 42 Let χ X h, then ε h (f, χ Ch ij f H i χ H j, f H i, i, j =, 1, ε a (R h v, χ Ch ij v H 1i χ H j, v H 1i H 1, i, j =, 1 Using the symmetry of (χ, I h ψ on X h, together with (111 and (23 we therefore obtain from (46 1 d 2 dt ϑ 2 a h (ϑ, I h ϑ C ϱ t ϑ Ch 2( u t H 1 f H 1 u H 2 ϑ H 1 (47 Ch 2( u t H 2 f H 1 ϑ H 1 Ch 2( u tt f t f H 1 ϑ H 1, t > Using (36 to kick back ϑ H 1, integrating, and using (31 we obtain, since ϑ( =, ϑ(t 2 Ch 4 ( u tt 2 f t 2 f 2 H dτ (48 1 To complete the estimation of ϑ we now only need the regularity estimate u tt 2 dτ C( g 2 H f 1 t 2 dτ, t T (49 A more general such estimate is derived in Evans [13, Chapter 7], for Ω smooth, using the method of Galerkin approximations In the special case we need, the regularity of Ω is not required Finally, by combining (42, (48, and (49, we obtain the desired estimate (41 V SOME OTHER ESTIMATES In this section we shall use our technique from Section IV, based on the standard Ritz projection, to derive error bounds also in H 1 and L norms We begin with the bound in H 1

13 FVEM ERROR ESTIMATES FOR PARABOLIC PDE S IN POLYGONS 13 Theorem 51 Let u h and u be the solutions of (15 and (11, respectively, and assume that u = on Ω and ε (, 1/2 Then, if u h = R hu, with R h defined in (11, we have where u h (t u(t H 1 C T,ε (u, fh, t T, (51 ( C T,ε (u, f = C T u H 2 g ( f t 2 dτ 1/2 ε 1 ( g H ε Proof In view of (111 we have ϱ(t H 1 Ch u(t H 2 Ch( u H 2 f t H ε dτ u t H 2 dτ, t (52 Using (113 this shows that ϱ(t is bounded as desired We now turn to the estimation of ϑ, using the identity (43 with η = I h ϑ t Since, a h (χ, I h ψ = a(χ, ψ ε a (χ, ψ then Thus, from (43 and (45 we get a h (ϑ, I h ϑ t = 1 d 2 dt a(ϑ, ϑ ε a(ϑ, ϑ t ϑ t 2 1 d 2 dt a(ϑ, ϑ = (ϱ t, I h ϑ t ε a (ϑ, ϑ t ε h (Lu, ϑ t ε a (R h u, ϑ t Substituting Lu = f u t this yields ϑ t 2 1 d 2 dt a(ϑ, ϑ = (ϱ t, I h ϑ t ε a (ϑ, ϑ t ε h (u t, ϑ t d dt ε h(f, ϑ ε h (f t, ϑ ε a (R h u, ϑ t (53 Integrating this relation over [, t], t T, using the fact that ϑ( =, together with (111, Lemma 42, (31, and (23, we find ϑ(t 2 H 1 Ch 2( f(t 2 ( u t 2 H 1 u 2 H 2 f t 2 dτ C Using Gronwall s lemma and the simple estimate ( u(t 2 H C( u t(t 2 f(t 2 C u 2 t (t 2 f( 2 now gives ϑ(t 2 H 1 Ch2( f( 2 Similarly to (49, one can show u t 2 H 1 dτ C( g 2 ϑ 2 H 1 dτ f t 2 dτ, ( u t 2 H 1 f t 2 dτ, t T (54 f t 2 dτ, t T, (55 by slightly modifying the proof of the corresponding result in [13] Combined with (54, this shows the desired bound for ϑ(t Together with (52 this completes the proof

14 14 P CHATZIPANTELIDIS, R D LAZAROV AND V THOMÉE We now turn to some error estimates in the maximum-norm Writing as usual u h u = (u h R h u (R h u u = ϑ ϱ we will show ϑ L = O(h 2 (log 1 h 1/2 At a corner the solution normally has a singularity, the strength of which depends on the angle at the corner Below we show that ϱ L = O(h γ for any γ < β = π/ω, cf (118, and since β > 1 we can take γ > 1 as well In our first result below we show a global maximum-norm error estimate of this order This will be done using an analysis of Schatz [22] for the elliptic problem in a polygonal domain We remark that when the domain has a smooth boundary it is known that ϱ L = O(h 2 (log 1 h 2, see [14, Chapter 5, Lemma 56] The loss of order of convergence only takes place near the corners, and in second result below we show a bound of the same order on domains Ω Ω, with Ω not containing any vertex of Ω We now state and show the global error estimate Theorem 52 Let u h and u be the solutions of (15 and (11, respectively, and assume u = g = on Ω, and that the family of triangulations {T h } <h<1 is quasi uniform Then if u h = R hu we have, for any γ < β, where u h (t u(t L C T,γ (u, fh γ, t T, (56 C T,γ (u, f = C ( T,γ u H 2 g L g H 1 g 1 f( L Proof We begin by showing that for ϱ = R h u u we have f t L dτ ( ( f t 2 H f 1 tt 2 dτ 1/2 ϱ(t L C γ h γ( g L f( L f t L dτ (57 We shall assume that 1 < γ < β < 2; the proof may easily be modified to cover the case β 2 Letting δ = (β γ/2 we have in view of [22, (8] that ϱ(t L Ch δ log 1 h u(t C 1,δ 1 Chγ u(t C 1,δ 1, where C 1,α is the space of continuously differentiable functions whose first order derivative fulfill a uniform Hölder condition of order α A standard imbedding argument, cf, eg, [15, Theorem 1452], shows u(t C 1,δ 1 C γ u(t W 2 p, with p = 2/(2 δ, and since p < 2/(2 β, we have the elliptic regularity estimate, cf, eg, [15, Theorem 527], u W 2 p C Lu Lp (58 Combining (27 and E(t L 1, we obtain ϱ(t L C γ h γ Lu L C γ h γ( u t (t L f(t L C γ h γ( g L f( L f t L dτ

15 FVEM ERROR ESTIMATES FOR PARABOLIC PDE S IN POLYGONS 15 As indicated above, in the case that Ω is smooth, we have ϱ L = O(h 2 (log 1 h 2 This is seen by using the estimates ϱ L C log 1 h I hu u L Ch 2 2/p log 1 h u W 2 p (59 where the error bound for the interpolant is easily obtained by the Bramble Hilbert lemma, cf, eg, [21, (4429], the elliptic regularity estimate u W 2 p Cp Lu Lp, 2 p <, (51 and choosing p = log 1 h, cf [14, Lemma 56] The estimate (51 does not hold for polygonal domains for p large, cf [15, Remark 4326] We now turn to the required estimate for ϑ = u h R h u We will show ϑ(t L Ch 2 (log 1 ( 1/2 u h H 2 g H 1 g 1 ( ( f t 2 H f 1 tt 2 dτ (511 1/2 In view of the well-known inequality, cf, eg, [14, Chapter 5], it suffices to demonstrate χ L C ( log 1 h 1/2 χ, χ Xh, ϑ(t H 1 Ch 2( u H 2 g H 1 g 1 ( ( f t 2 H f 1 tt 2 dτ 1/2 For this we rewrite (53 as ϑ t 2 1 d 2 dt a(ϑ, ϑ = (ϱ t, I h ϑ t ε a (ϑ, ϑ t d dt ε h(u t f, ϑ ε h (u tt f t, ϑ d dt ε a(r h u, ϑ ε a (R h u t, ϑ Since ϑ( =, (111, Lemma 42, (31, and (48 give, for t, ϑ(t 2 H 1 Ch 4( u t (t 2 H 1 u(t 2 H 2 f(t 2 H 1 ( u tt 2 H u 1 t 2 H f 2 t 2 H dτ C 1 ϑ 2 H 1 dτ, and by Gronwall s lemma and obvious estimates we now have, for t T, ϑ(t 2 H 1 Ch 4( u 2 H 2 g 2 H 1 To complete the proof we need the regularity estimate u tt 2 H 1 dτ C( g 1 2 which can be found in [13] in the same way as (49 ( u tt 2 H 1 f t 2 H 1 dτ f tt 2 dτ, t T,

16 16 P CHATZIPANTELIDIS, R D LAZAROV AND V THOMÉE We note that since we only require a O(h β estimate for ϑ L, the regularity assumptions used in this part of the proof could be reduced We now turn to the error away from the corners, and demonstrate that there we can improve the convergence rate to essentially second order Theorem 53 Let Ω Ω be such that Ω does not contain any vertex of Ω, and assume the family of triangulations {T h } <h<1 is quasi uniform Let u h and u be the solutions of (15 and (11, respectively, and assume u = g = on Ω, ε (, 1/2 and 2 < p < 2/(2 β Then, if u h = R hu, we have where u h (t u(t L (Ω C T,p,ε(u, fh 2 log 1, t T, (512 h C T,p,ε (u, f = C T,p ( u H 2 g H 1 g 1 f(t W 1 p f t ( f tt dτ ( ( f t 2 H f 1 tt 2 dτ 1/2 ε 1 f t H ε dτ Proof It follows from (511 in the proof of the previous theorem that ϑ L is bounded as stated It therefore suffices to consider ϱ, and for this we shall use the following lemma which is a special case of a result from a forthcoming paper of Schatz [23] Lemma 54 Let Ω be a convex polygonal domain in R 2 and let Ω Ω 1 Ω, be such that Ω 1 does not contain any corners of Ω and the distance between Ω 1 Ω and Ω Ω is positive Then, if v h X h satisfies we have, for any χ X h, a(v h v, χ =, χ X h, v h v L (Ω C( log 1 h v χ L (Ω 1 v h v L2 Let Ω 2 and Ω 3 be domains with Ω 1 Ω 2 Ω 3 Ω and smooth boundaries and such that Ω 3 does not contain any corner of Ω Assume further that the distances between Ω 3 Ω, Ω 2 Ω, and Ω 1 Ω are positive Let w be a smooth cutoff function such that w Ω2 = 1 and w Ω3 Ω = Let Īh : C(Ω X h, be the standard nodal interpolant Then in view of the interpolation error estimate Lemma 54 gives Īhu u L (Ω 1 Ch2 u W 2 (Ω 2, ρ(t L (Ω Ch2 log 1 h u(t W 2 (Ω 2 C ρ(t Since the last term is bounded as desired by (42 it now remains to bound the first term on the right Using first a Sobolev inequality and then an elliptic regularity estimate in Ω 3 (recalling that Ω 3 is smooth, and, finally, the elliptic regularity estimate (58 in Ω,

17 FVEM ERROR ESTIMATES FOR PARABOLIC PDE S IN POLYGONS 17 we have, with ũ = wu, u W 2 (Ω 2 C u W 3 p (Ω 2 C ũ W 3 p (Ω 3 C Lũ W 1 p (Ω 3 C( Lu W 1 p u W 2 p C Lu W 1 p Using the differential equation, and then again a Sobolev inequality, followed by (23, we find Lu W 1 p C ( f W 1 p u t W 1 p C ( f W 1 p u t H 2 C ( f W 1 p u tt f t By (39, with E(t 1, we hence find u(t W 2 (Ω 2 C ( f(t W 1 p g 1 f t ( f tt dτ, which completes the proof of the desired estimate for ρ and hence of the theorem VI THE LUMPED MASS FINITE VOLUME METHOD In addition to the finite volume method (15 studied above we consider also a lumped mass variant of this method, which we define by approximating the first integral in (14 by means of a simple quadrature formula We thus seek u h (t X h satisfying u h,t (z, t V z (A u h n ds = f dx, z Zh, t, V z V z (61 u h ( = u h Note that this discrete scheme is no longer locally conservative In order to rewrite (61 in variational form we consider the quadrature formula Q K,h (f = 1 3 K f(z f dx, for K T h, z Z h (K and the associated bilinear form in X h Y h, (χ, ψ h = Q K,h (χψ = χ(zψ(z V z, χ X h, ψ Y h K T h z Z h We note that χ h := (χ, I h χ 1/2 h is a norm in X h which is equivalent with the L 2 norm, uniformly in h, ie, with c >, K c χ χ h C χ, χ X h (62 For an arbitrary η Y h, by multiplying (61 by η(z and summing over all z Z h, we obtain the Petrov-Galerkin formulation (u h,t, η h a h (u h, η = (f, η, η Y h, t (63 For this analogue of (15 the mass matrix M in (18 is replaced by the diagonal matrix M = (m zz where m zz = Φ z 2 h The name lumped mass method is inherited

18 18 P CHATZIPANTELIDIS, R D LAZAROV AND V THOMÉE from its analogue for the finite element equation (12 and is motivated by the fact that with W z = supp Φ z and V z = supp Ψ z we have, except when z is a neighbor of a boundary vertex, W z w V w and hence, cf also (32 and (33, m zw = (Φ z, Ψ w = Φ z dx = V z = m zz w Zh w wv w When z has a neighbor on Ω we have m zz > w Z h m zw We will now show that the L 2 norm error bound of Theorem 41 remains valid for the lumped mass method (63 Theorem 61 Let u h and u be the solutions of (63 and (11, respectively, and assume u = g = on Ω, and ε (, 1/2 Then, if u h = R hu, the error bound (41 holds Proof In view of our earlier estimate (42 for ρ = R h u u, it suffices to bound ϑ = u h R h u, which now satisfies (ϑ t, η h a h (ϑ, η = (ϱ t, η a h (ϱ, η ε h (R h u t, η, η Y h, (64 where ε h (, is the quadrature error defined by ε h (χ, η = (χ, η (χ, η h, χ X h, η Y h Note that the equation (64 differs from (43 only in the first term on the left and in the addition of the last one on the right We note that the above definition of (χ, ψ h may be used also for ψ X h, and that (χ, ψ h = (χ, I h ψ h for χ, ψ X h We therefore have, using the notation (44, It follows from, eg, [14, Lemma 151], that ε h (χ, I h ψ = ε h (χ, ψ ( (χ, ψ (χ, ψ h (χ, ψ h (χ, ψ Ch 2 χ H 1 ψ H 1, χ, ψ X h and hence, using also Lemma 41, that ε h (R h u t, I h ϑ Ch 2 u t H 1 ϑ H 1 (65 Therefore, setting η = I h ϑ in (64, and using (62 and (65, we obtain (48 as before, which shows the theorem We remark that the error bounds of Theorems 51, 52, and 53 also hold for the solution u h of (63 In fact, the estimates for ϱ in the proofs of these theorems remain unchanged, and the bounds for ϑ in Section V may be shown from (64 similarly to the corresponding bounds in Theorem 61 As in Theorem 31, instead of using R h we could use R h to obtain an L 2 norm error estimate for the solution u h of (63 In such a case ϑ = R h u u would satisfy ( ϑ t, η h a h ( ϑ, η = ( ϱ t, η ε h ( R h u t, η, η Y h and, in view of (62, (65, and (117 we would now obtain ϑ(t 2 Ch 4( ( u tt 2 W f t 2 p 1 W dτ, for p < p p 1

19 FVEM ERROR ESTIMATES FOR PARABOLIC PDE S IN POLYGONS 19 However, since stronger assumptions on f t than those in (48 are required to bound the integral, we shall not elaborate on this choice We recall that with the above notation the lumped mass finite element method for (11 is (u h,t, χ h a(u h, χ = (f, χ, χ X h, t (66 We remark that in the case of a constant coefficient matrix A, the left sides of (66 and (63 with η = I h χ are the same, cf (35, so that the two lumped mass methods differ only in the treatment of the inhomogeneous term VII FULLY DISCRETE METHODS In this section, for completeness, we will show by an example that our approach to the finite volume method applies also to fully discrete schemes We consider the backward Euler method for the discretization in time of (15 Letting k be the time step, U n the approximation in X h of u(t at t = t n = nk, and U n = (U n U n 1 /k, this method is defined by ( U n, η a h (U n, η = (f(t n, η, η Y h, n 1, U = u h We then have the following error estimate (71 Theorem 71 Let U n and u(t n be the solutions of (71 and (11, respectively, and assume u = g = on Ω and ε (, 1/2 Then, if U = R h u, we have where and U n u(t n C T,ε (u, fh 2 C T (u, fk, t n T, (72 C T,ε (u, f = C T ( u H 2 g H 1 ε 1 n Proof We write C T (u, f = C T ( g H 1 ( n f t H ε dτ ( k f t 2 dτ 1 2 n j= U n u(t n = (U n R h u(t n (R h u(t n u(t n ϑ n ϱ n f j 2 1 H 1 2 and since the estimate for ρ n = ρ(t n is the same as before, it suffices to bound ϑ n We have ( ϑ n, η a h (ϑ n, η = ( ϱ n, η a h (ϱ n, η (u n t u n, η, η Y h Choosing η = I h ϑ n we obtain, with defined in (31, cf (46, 1 2k ( ϑn 2 ϑ n k ϑn ϑ n 1 2 a h (ϑ n, I h ϑ n = ( ϱ n, I h ϑ n ε h (u n t f n, ϑ n ε a (R h u n, ϑ n (u n t u n, I h ϑ n = ( ϱ n, I h ϑ n ε h ( u n f n, ϑ n ε a (R h u n, ϑ n (u n t u n, ϑ n

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