summary:In this paper we study the finite element approximations to the Sobolev and viscoelasticity type equations and present a direct analysis for global superconvergence for these problems, without using Ritz projection or its modified forms

Lin, Qun

Zhang, Shuhua

English

Institute of Mathematics AS CR, v. v. i.

Applications of MathematicsQun Lin; Shu Hua ZhangA direct global superconvergence analysis for Sobolev and viscoelasticity typeequationsApplications of Mathematics, Vol. 42 (1997), No. 1, 23–34Persistent URL: http://dml.cz/dmlcz/134342Terms of use:© Institute of Mathematics AS CR, 1997Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This document has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://dml.cz42 (1997) APPLICATIONS OF MATHEMATICS No. 1, 23–34A DIRECT GLOBAL SUPERCONVERGENCE ANALYSISFOR SOBOLEV AND VISCOELASTICITY TYPE EQUATIONSQun Lin, Shuhua Zhang, Beijing(Received June 16, 1995)Abstract. In this paper we study the finite element approximations to the Sobolev andviscoelasticity type equations and present a direct analysis for global superconvergence forthese problems, without using Ritz projection or its modified forms.Keywords: Sobolev and viscoelasticity type equations, global superconvergence, directanalysisMSC 2000 : 65B05, 65N301. IntroductionLet Ω be a rectangular domain. In order to explain our superconvergence analysisfor FEMs succinctly, we only consider the simple Sobolev type equation(1.1)−ut −u = f in Ω× (0, T ],u = 0 on ∂Ω× (0, T ],u(x, 0) = v in Ω,and a viscoelasticity type equation(1.2)utt −ut −u = f in Ω× (0, T ],u = 0 on ∂Ω× (0, T ],u(x, 0) = v, ut(x, 0) = w in Ω.In (1.1) and (1.2), f , v and w are sufficiently smooth functions.The problems (1.1) and (1.2) can arise from many physical processes. The nu-merical approximations to the solutions of these problems have been investigated23by Ewing [2, 3], Ford [4], Ford and Ting [5, 6], and Wahlbin [14]. Also Arnold,Douglas and Thomée [1] and Nakao [13] have considered Galerkin approximations tothe soluton of the problem (1.1) in a single space dimension with periodic boundaryconditions. L2-error estimates and the interior pointwise superconvergence resultshave been derived by these authors. In particular, some futher investigations of thefinite element methods for the problems (1.1) and (1.2) have been carried out by Lin,Thomée and Wahlbin [12].According to the conventional error analysis for FEMs of the time-dependent prob-lems, either the Ritz projection initiated by Wheeler [15] or its modifided formse.g. the so-called Ritz-Volterra projection introduced by Lin et al. [12], have to beused as transitional tools. However, here we will use a new analysis in [7], i.e. ananalysis for the “short side” in the FE-right triangle plus the sharp integral esti-mates of the “hypotenuse”, instead of using Ritz projection or its modified forms,to gain the global superconvergence for the problems (1.1) and (1.2) by an interpo-lation postprocessing technique, rather than the interior pointwise superconvergenceby means of the average technique with which numerical analysts are familiar. Ouranalysis sharpens the results and shortens the proofs of error estimates that appearedin the previous literature under the rectangular mesh assumption.2. Sobolev type equationsFirst of all, we discuss the model problem (1.1). Here and below, assume that T his a rectangular partition over Ω with mesh size h. The weak form of (1.1) consistsin finding u(. , t) ∈ H10 (Ω) for any t ∈ [0, T ] (the Sobolev space) such that(2.1){(∇ut,∇ϕ) + (∇u,∇ϕ) = (f, ϕ) ∀ϕ ∈ H10 ,u(0) = v.Let Sh0 ⊂ H10 consist of piecewise bilinear functions. Thus, a continuous Galerkinapproximation uh(x, t) : [0, T ]→ Sh0 is defined so that(2.2){(uht ,ϕ) + (uh,ϕ) = (f, ϕ) ∀ϕ ∈ Sh0 ,uh(0) = ihv,where ihv ∈ Sh0 is the bilinear interpolation function of v. From (2.1) and (2.2) weget the error equation(2.3) (∇(uht − ut),∇ϕ) + (∇(uh − u),∇ϕ) = 0 ∀ϕ ∈ Sh0 .We need the following (see [9])242.1. Lemma. For ϕ ∈ Sh0 ,|(∇(u − ihu),∇ϕ)| {ch2‖u‖3|ϕ|1,ch2‖u |4‖ϕ‖0.2.1. Theorem. For sufficiently smooth u and ut, we have‖uh − ihu‖1  ch2[∫ t0(‖ut‖3 + ‖u‖3)2 ds]1/2. . Letθ(x, t) = uh − ihu.Then, by virtue of (2.3) we have for ϕ ∈ Sh0(2.4) (∇θt,∇ϕ) + (∇θ,∇ϕ) = (∇(ut − ihut),∇ϕ) + (∇(u − ihu),∇ϕ).Hence, with ϕ = θt (where θt is the classical derivative) and Lemma 2.1,|θt|21 +12ddt|θ|21  ch2(‖ut‖3 + ‖u‖3)|θt|1or|θt|21 +12ddt|θ|21  ch4(‖ut‖3 + ‖u‖3)2 +12|θt|21.By integration with respect to t, it follows from θ(x, 0) = 0 that|θ|1  ch2[∫ t0(‖ut‖3 + ‖u‖3)2 ds]1/2.In order to derive L∞ estimates, we introduce the discrete Green function Ghz ∈ Sh0at any point z∈ Ω such that, for ϕ ∈ Sh0 ,(∇Ghz ,∇ϕ) = ϕ(z), (∇DzGhz ,∇ϕ) = Dzϕ.We need the following2.2. Lemma. ([16]) ‖Ghz‖0  c, ‖DzGhz‖0  c(log1h)1/2.252.2. Theorem. For sufficiently smooth u and ut, we have‖uht − ut‖0,∞  ch2(‖ut‖2,∞ + ‖ut‖4 + ‖u‖4). . Taking ϕ = Ghz in (2.4), we have by Lemma 2.1 and Lemma 2.2(2.5) θt(z, t) + θ(z, t)  ch2(‖ut‖4 + ‖u‖4),and hence, with θ(0) = 0,θt(z, t) +∫ t0θt(z, s) ds  ch2(‖ut‖4 + ‖u‖4).Therefore, it follows from Gronwall’s Lemma that|θt(z, t)|  ch2(‖ut‖4 + ‖u‖4).Thus, Theorem 2.2 holds in virtue of the triangle inequality‖uht − ut‖0,∞  ‖uht − ihut‖0,∞ + ‖ihut − ut‖0,∞.2.3. Theorem. For sufficiently smooth u and ut, we have‖uh − u‖0,∞  ch2(‖u‖2,∞ + ‖u‖4 + ‖ut‖4). . From (2.5) and Theorem 2.2, we drive|θ(z, t)|  ch2(‖ut‖4 + ‖u‖4),and hence, Theorem 2.3 holds in virtue of the triangle inequality‖uh − u‖0,∞  ‖uh − ihu‖0,∞ + ‖ihu− u‖0,∞.262.4. Theorem. For sufficiently smooth u and ut, we have‖uht − ihut‖1,∞  ch2(log1h)1/2(‖ut‖4 + ‖u‖4). . Setting ϕ = DzGhz in (2.4), we get by means of Lemma 2.1 and 2.2(2.6) Dzθt(z, t) +Dzθ(z, t)  ch2(log1h)1/2(‖ut‖4 + ‖u‖4).And thus, with Dzθ(0) = 0,Dzθt(z, t) +∫ t0Dzθt(z, s) ds  ch2(log1h)1/2(‖ut‖4 + ‖u‖4).Then, Theorem 2.4 follows from Gronwall’s Lemma. 2.5. Theorem. For sufficiently smooth u and ut, we have‖uh − ihu‖1,∞  ch2(log1h)1/2(‖ut‖4 + ‖u‖4). . From (2.6) and Theorem 2.4 we obtain Theorem 2.5. Theorems 2.1, 2.4 and 2.5 play key roles in the analysis of the global superconver-gence for the problem (1.1). Now we use an interpolation postprocessing techniquefrom [8] to get the desired results. We assume that T h has been gained from T 2hwith mesh size 2h by subdividing each element of T 2h into four congruent elements.Thus, we can define a nodal biquadratic interpolation operator I22h associated withT 2h. It is easy to check thatI22hih = I22h, ‖I22hϕ‖1,p  c‖ϕ‖1,p ∀ϕ ∈ Sh0 (p = 2,∞),‖I22hϕ− ϕ‖1,p  ch2‖ϕ‖3,p (p = 2,∞).And thus, we have the following main results.2.6. Theorem. For sufficiently smooth u and ut, we have‖I22huh − u‖1  ch2{‖u‖3 +[∫ t0(‖ut‖3 + ‖u‖3)2 ds]1/2}.27 . Due to the property of I22h, we haveI22huh − u = I22h(uh − ihu) + (I22hu− u).Therefore, it follows from Theorem 2.1 and the interpolation theorem that‖I22huh − u‖1  c‖uh − ihu‖1 + ch2‖u‖3 ch2{‖u‖3 +[∫ t0(‖ut‖3 + ‖u‖3)2 ds]1/2}.Analogously, by Theorems 2.4 and 2.5 we have the next assertions.2.7. Theorem. For sufficiently smooth u and ut, we have‖I22huht − ut‖1,∞  ch2(log1h)1/2(‖ut‖3,∞ + ‖ut‖4 + ‖u‖4).2.8. Theorem. For sufficiently smooth u and ut, we have‖I22huh − u‖1,∞  ch2(log1h)1/2(‖u‖3,∞ + ‖ut‖4 + ‖u‖4).3. Viscoelasticity type equationsIn this section, we will consider the semidiscrete Galerkin approximation to theproblem (1.2). The weak form of (1.2) reads as follows: Find u(. , t) ∈ H10 (Ω) for anyfixed t ∈ [0, T ] such that(3.1){(utt, ϕ) + (∇ut,∇ϕ) + (∇u,∇ϕ) = (f, ϕ) ∀ϕ ∈ H10 (Ω),u(0) = v, ut(0) = w.Thus, a continuous Galerkin approximation uh(x, t) : [0, T ]→ Sh0 is defined so that(3.2){(uhtt, ϕ) + (∇uht ,∇ϕ) + (∇uh,∇ϕ) = (f, ϕ) ∀ϕ ∈ Sh0 ,uh(0) = ihv, uht (0) = ihw,where ihv, ihw ∈ Sh0 stand for the bilinear interpolation functions of v and w,respectively. Then, we obtain the error equation from (3.1) and (3.2):(3.3) (utt − uhtt, ϕ) + (∇(ut − uht ),∇ϕ) + (∇(u− uh),∇ϕ) = 0 ∀ϕ ∈ Sh0 .283.1. Theorem. For sufficiently smooth u, ut and utt, we have‖uh − ihu‖1 + ‖uht − ut‖0  ch2{‖ut‖2 +[∫ t0(‖utt‖2 + ‖ut‖4 + ‖u‖4)2 ds]1/2}. . Letθ(x, t) = uh(x, t)− ihu(x, t).According to (3.3), we have for ϕ ∈ Sh0 that(3.4)(θtt, ϕ) + (∇θt,∇ϕ) + (∇θ,∇ϕ) = (utt − ihutt, ϕ) + (∇(ut − ihut),∇ϕ)+ (∇(u − ihu),∇ϕ),and hence, with ϕ = θt and Lemma 2.1,12ddt‖θt‖20 + c‖θt‖21 +12ddt|θ|21 12ddt‖θt‖20 + |θt|21 +12ddt|θ|21 ch4(‖utt‖2 + ‖ut‖4 + ‖u‖4)2 +c2‖θt‖20,orddt(‖θt‖20 + |θ|21)  ch4(‖utt‖2 + ‖ut‖4 + ‖u‖4)2.By integration with respect to t, it follows from θ(0) = θt(0) = 0 that‖θt‖20 + |θ|21  ch4t∫0(‖utt‖2 + ‖ut‖4 + ‖u‖4)2 ds,that is‖θt‖0 + |θ|1  ch2[∫ t0(‖utt‖2 + ‖ut‖4 + ‖u‖4)2 ds]1/2.And thus, Theorem 3.1 follows from the triangle inequality‖uht − ut‖0  ‖uht − ihut‖0 + ‖ihut − ut‖0.3.2. Theorem. For sufficiently smooth u, ut, utt and uttt, we have‖uht − ihut‖1 + ‖uhtt − utt‖0  ch2{‖utt‖2 + [(‖utt(0)‖2 + ‖ut(0)‖4 + ‖u(0)‖4)2+∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds]1/2}.29 . Differentiating (3.4) with respect to t, we get, for ϕ ∈ Sh0 ,(θttt, ϕ) + (∇θtt,∇ϕ) + (∇θt,∇ϕ) = (uttt − ihuttt, ϕ) + (∇(utt − ihutt),∇ϕ)+ (∇(ut − ihut),∇ϕ),and hence, with ϕ = θtt and Lemma 2.1,12ddt‖θtt‖20 + c‖θtt‖21 +12ddt|θt|21 12ddt‖θtt‖20 + |θtt|21 +12ddt|θt|21 ch4(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 +c2‖θtt‖20,orddt(‖θtt‖20 + |θt|21)  ch4(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2.Therefore, by virtue of θt(0) = 0, we get(3.5) ‖θtt‖20 + |θt|21  ‖θtt(0)‖20 + ch4∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds.Let t = 0 and ϕ = θtt(0) in (3.4). Then‖θtt(0)‖0  ch2(‖utt(0)‖2 + ‖ut(0)‖4 + ‖u(0)‖4)which, together with (3.5), leads to‖θtt‖20 + |θt|21  ch4[(‖utt(0)‖2 + ‖ut(0)‖4 + ‖u(0)‖4)2+∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds],or‖θtt‖0 + ‖θt‖1  ch2[(‖utt(0)‖2 + ‖ut(0)‖4 + ‖u(0)‖4)2+∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds]1/2,and Theorem 3.2 follows from the triangle inequality‖uhtt − utt‖0  ‖uhtt − ihutt‖0 + ‖ihutt − utt‖0.303.3. Theorem. For sufficiently smooth u, ut, utt and uttt, we have‖uht − ut‖0,∞  ch2{‖utt‖2 + ‖ut‖4 + ‖ut‖2,∞ + ‖u‖4+ [(‖utt(0)‖2 + ‖ut(0)‖4 + ‖u(0)‖4)2+∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds]1/2}. . Setting ϕ = Ghz in (3.4), we have according to Lemmas 2.1 and 2.2(3.6) (θtt, Ghz ) + θt(z, t) + θ(z, t)  ch2(‖utt‖2 + ‖ut‖4 + ‖u‖4),or|θt(z, t)| ∫ t0|θt(z, s)| ds+ c‖θtt‖0 + ch2(‖utt‖2 + ‖ut‖4 + ‖u‖4).And thus, it follows from Gronwall’s Lemma and Theorem 3.2 that|θt(z, t)|  ch2{‖utt‖2 + ‖ut‖4 + ‖u‖4+ [(‖utt(0)‖2 + ‖ut(0)‖4 + ‖u(0)‖4)2+∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds]1/2}.Then, Theorem 3.3 holds by virtue of the triangle inequality‖uht − ut‖0,∞  ‖uht − ihut‖0,∞ + ‖ihut − ut‖0,∞.3.4. Theorem. For sufficiently smooth u, ut, utt and uttt, we have‖uh − u‖0,∞  ch2{‖u‖2,∞ + ‖u‖4 ++‖ut‖4 + ‖utt‖2+ [(‖utt(0)‖2 + ‖ut(0)‖4 + ‖u(0)‖4)2+∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds]1/2}. . From (3.6), Lemma 2.2 and Theorems 3.2, 3.3 we derive|θ(z, t)|  |θt(z, t)|+ c‖θtt‖0 + ch2(‖utt‖2 + ‖ut‖4 + ‖u‖4) ch2{‖utt‖2 + ‖ut‖4 + ‖u‖4 + [(‖utt(0)‖2 + ‖ut(0)‖4+ ‖u(0)‖4)2 +∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds]1/2},31and hence, Theorem 3.4 holds in virtue of the triangle ineqality‖uh − u‖0,∞  ‖uh − ihu‖0,∞ + ‖ihu− u‖0,∞.3.5. Theorem. For sufficiently smooth u, ut, utt and uttt, we have‖uht − ihut‖1,∞  ch2(log1h)1/2{‖utt‖2 + ‖ut‖4 + ‖u‖4+ [(‖utt(0)‖2 + ‖ut(0)‖4 + ‖u(0)‖4)2+∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds]1/2}. . Taking ϕ = DzGhz in (3.4), we obtain according to Lemmas 2.1 and 2.2(3.7) (θtt, DzGhz ) +Dzθt(z, t) +Dzθ(z, t)  ch2(log1h)1/2(‖utt‖2 + ‖ut‖4 + ‖u‖4),or|Dzθt(z, t)| ∫ t0|Dzθt(z, s)| ds+ c(log1h)1/2‖θtt‖0+ ch2(log1h)1/2(‖utt‖2 + ‖ut‖4 + ‖u‖4).And thus, Theorem 3.5 follows from Gronwall’s Lemma and Theorem 3.2. 3.6. Theorem. For sufficiently smooth u, ut, utt and uttt, we have‖uh − ihu‖1,∞  ch2(log1h)1/2{‖utt‖2 + ‖ut‖4 + ‖u‖4+ [(‖utt(0)‖2 + ‖ut(0)‖4 + ‖u(0)‖4)2+∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds]1/2}. . From (3.7) and Theorems 3.2, 3.5 we obtain Theorem 3.6. Theorems 3.1, 3.2, 3.5 and 3.6 are essential, by them we can get the global super-convergence for the problem (1.2) instead of the interior pointwise superconvergence.Identically to Section 2, we derive the following main theorems by means of the in-terpolation postprocessing technique initiated in [8].323.7. Theorem. For sufficiently smooth u, ut and utt, we have‖I22huh − u‖1  ch2{‖ut‖2 + ‖u‖3 +[∫ t0(‖utt‖2 + ‖ut‖4 + ‖u‖4)2 ds]1/2}.3.8. Theorem. For sufficiently smooth u, ut, utt and uttt, we have‖I22huht − ut‖1  ch2{‖utt‖2 + ‖ut‖3+ [(‖utt(0)‖2 + ‖ut(0)‖4 + ‖u(0)‖4)2+∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds]1/2}.3.9. Theorem. For sufficiently smooth u, ut, utt and uttt, we have‖I22huht − ut‖1,∞  ch2(log1h)1/2{‖utt‖2 + ‖ut‖4 + ‖ut‖3,∞+ ‖u‖4 + [(‖utt(0)‖2 + ‖ut(0)‖4 + ‖u(0)‖4)2+∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds]1/2}.3.10. Theorem. For sufficiently smooth u, ut, utt and uttt, we have‖I22huh − u‖1,∞  ch2(log1h)1/2{‖utt‖2 + ‖ut‖4 + ‖u‖3,∞+ ‖u‖4 + [(‖utt(0)‖2 + ‖ut(0)‖4 + ‖u(0)‖4)2+∫ t0(‖uttt‖2 + ‖utt‖4 + ‖ut‖4)2 ds]1/2}. 1. In another paper, we will discuss the case of k (k  2) which is theorder of finite elements for the problems above. 2. When Ω is a convex quadrilateral domain, the corresponding super-convergent results hold for such problems as above if the quadrilateral meshes arealmost uniform and are constructed by connecting the equi-proportional points oftwo opposite boundaries.33Acknowledgement. Prof. Y.P. Lin generously gave us his papers about Sobolevand viscoelasticity equations when the first author visited Canada in 1994. His workin this field has aroused our interest in such problems. The authors would like tothank Prof. M. Křížek whose comments improved this version of the paper.References[1] D. Arnold, J. Douglas, V. Thomée: Superconvergence of a finite element approximationto the solution of a Sobolev equation in a single space variable. Math. Comp. 36 (1981),53–63.[2] R. Ewing: The approximation of certain parabolic equations backward in time bySobolev equations. SIAM J. Math. Anal. 6 (1975), 283–294.[3] R. Ewing: Numerical solution of Sobolev partial differential eqautions. SIAM J. Numer.Anal. 12 (1975), 345–363.[4] W. Ford: Galerkin approximation to nonlinear pseudoparabolic partial differential equa-tion. Aequationes Math. 14 (1976), 271–291.[5] W. Ford, T. Ting: Stability and convergence of difference approximations to pseudopar-abolic partial equations. Math. Comp. 27 (1973), 737–743.[6] W. Ford, T. Ting: Uniform error estimates for difference approximations to nonlin-ear pseudoparabolic partial differential equations. SIAM J. Numer. Anal. 11 (1974),155–169.[7] Q. Lin: A new observation in FEM. Proc. Syst. Sci. & Syst. Eng. (1991), 389–391. GreatWall (H.K.) Culture Publish Co..[8] Q. Lin, N. Yan, A. Zhou: A rectangle test for interpolated finite elements, ibid.[9] Q. Lin, S. Zhang: An immediate analysis for global superconvergence for integrodiffer-ential equations. Appl. Math. 42 (1997), 1–21.[10] Y. Lin: Galerkin methods for nonlinear Sobolev equations. Aequations Math. 40 (1990),54–56.[11] Y. Lin, T. Zhang: Finite element methods for nonlinear Sobolev equations with nonlin-ear boundary conditions. J. Math. Anal. & Appl. 165 (1992), 180–191.[12] Y. Lin, V. Thomée, L. Wahlbin: Ritz-Volterra projection on finite element spaces andapplications to integrodifferential and related equations. SIAM J. Numer. Anal. 28(1991), 1047–1070.[13] M. Nakao: Error estimates of a Galerkin method for some nonlinear Sobolev equationsin one space dimension. Numer. Math. 47 (1985), 139–157.[14] L. Wahlbin: Error estimates for a Galerkin method for a class of model equations forlong waves. Numer. Math. 23 (1975), 289–303.[15] M. Wheeler: A priori L2 error estimates for Galerkin approximations to parabolic partialdifferential equations. SIAM J. Numer. Anal. 10 (1973), 723–759.[16] Q. Zhu, Q. Lin: Superconvergence Theory of the Finite Element Methods. Hunan Sci-ence Press, 1990.Author’s address: Qun Lin, Shuhua Zhang, Institute of Systems Science, AcademiaSinica, Beijing 100080, P. R. China.34

A direct global superconvergence analysis for Sobolev and viscoelasticity type equations

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A direct global superconvergence analysis for Sobolev and viscoelasticity type equations

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