Research Article | Open Access
Strong Superconvergence of Finite Element Methods for Linear Parabolic Problems
We study the strong superconvergence of a semidiscrete finite element scheme for linear parabolic problems on , where is a bounded domain in with piecewise smooth boundary. We establish the global two order superconvergence results for the error between the approximate solution and the Ritz projection of the exact solution of our model problem in and with and the almost two order superconvergence in and . Results of the case are also included in two space dimensions ( or 2). By applying the interpolated postprocessing technique, similar results are also obtained on the error between the interpolation of the approximate solution and the exact solution.
Consider the followinginitial boundary value parabolic problem: where is a bounded domain in with piecewise smooth boundary , and is a second-order symmetric positive definite elliptic operator. Coefficients of and together with their derivatives up to certain order are bounded in order to guarantee our analysis. Note that our assumptions on do not have any restrictions, since it will be shown that approximate solutions considered below are uniformly close to the exact solution and thus only depend on the data of (1.1) in a neighborhood of .
Superconvergence of finite element methods for parabolic problems has been studied in many works. For example, Thomée , Chen and Huang  studied superconvergence of the gradient in norm. In 1989, Thomée et al.  studied maximum-norm superconvergence of the gradient in piecewise linear finite element approximations of a parabolic problem. An analogous result was also obtained by Chen . Moreover, Li and Wei  investigated global strong superconvergence of finite element schemes for a class of Sobolev equations in , and two order superconvergence results are proved in and for . In particular, Kwak et al.  studied superconvergence of a semi-discrete finite element scheme for parabolic problems in , in which superconvergence results in and are established for .
In this paper, we extend superconvergence results obtained in . We derive the two order and the almost two order global superconvergence estimates of in and in , where is the approximate solution, and is the Ritz projection of the exact solution of (1.1). In addition to the results in , we establish two order superconvergence estimates in norm for piecewise cubic or higher elements. Moreover, results of the case are also included in two space dimensions or . As an application, by employing the interpolated finite element operators (cf. [7, 8]) to the approximate solution in the rectangular mesh, we obtain the two order global superconvergence of the error between and the interpolation of . For a general domain , we can also apply the optimal partition to most rectangular meshes to derive one and a half- order superconvergence.
The rest of this paper is organized as follows. Section 2 provides some preliminaries. Several useful lemmas are established in Section 3. In Sections 4 and 5, we derive the superconvergence in and respectively. Finally, an application is presented in Section 6.
We denote and , and , the Sobolev spaces on associated with the norms
If is a normed space with the norm and , then we define
Moreover, we denote the inner product in (or ) and (or ), respectively. We also use Sobolev spaces with norm
We use to denote a generic positive constant independent of that can take different values at different occurrences.
Lemma 2.1. For all and , we have
Lemma 2.2. For all , we have
Given a function , we define its Ritz projection that satisfies Then we get the following well-known estimate:
Moreover, by using the duality argument and (2.9), it is true that for , where and .
We now turn to the finite element scheme of (1.1).
Find a map such that where is defined by and is given by (1.1).
3. Auxiliary Lemmas
To investigate the superconvergence of finite element approaches for parabolic problems, here and throughout the paper, we decompose the error as and estimate in a superconvergent order.
We start with the superconvergence of initial value errors.
Proof. From (2.12) and (2.11), we have for all
and thus (3.1) holds.
By the definition of , (2.11), (2.8), (1.1), and the definition of , we obtain that for all , Let , and note that , we obtain Then by taking , it follows from (2.10) that which implies (3.2).
Finally we turn to the proof of (3.3). To do so, we construct an auxiliary problem. Let satisfy and hence by the regularity estimate, it holds that where .
Therefore, it follows from (3.8), (2.8), (3.6), (2.10), (2.9), and (3.9) that for , which implies (3.3).
The following lemma gives superconvergence estimates for and .
Proof. By differentiating (3.5) in time, we have Choosing , (3.12) becomes Integrating both sides of (3.13) with respect to and applying the integration by parts argument, we obtain that from (3.1) and (2.10) Therefore, the proof is completed by eliminating and applying the Gronwall inequality.
Lemma 3.3. It holds that
4. Superconvergence in
In this Section, we derive the two order global superconvergence () and the almost two order global superconvergence () estimates on in .
Theorem 4.1. Under the assumptions that and , we have for and ,
Proof. We first introduce an auxiliary problem.
For , let be an arbitrary component of , and let be the solution of The following priori estimate holds:
Let in (4.2), it follows from the integration by parts argument, (2.8) and (3.5) that
From (2.10), the stability of and (4.3), we obtain
On the other hand, for and or , or , and or , Sobolev embedding inequalities (cf. ), Lemma 3.2, the stability of and (4.3) imply that
Combining (4.4), (4.5), and (4.6), we have
Therefore, (4.1) follows from summing up all components of .
We now turn to the case of .
Theorem 4.3. Assume that and . Then for and , where is large enough and
Proof. We first define the Green functions associated with the bilinear form .
Let be the pre-Green function, and let be the directional derivative of along some direction with respect to . Let be the finite element approximations of and , respectively. Then we know that (cf. [1, 12])
Now by definitions of Green functions, (3.5), Hölder's inequalities, (2.10), and (3.11), it is true that for all , which together with (4.10) yields
Similarly, by the inverse property (2.6) and (4.11), we have which implies that, for large enough and sufficiently small,
Inequality (4.9) then follows from (4.13) and (4.15).
5. Superconvergence in
In this section, we establish the strong superconvergence for in with .
We start with the following two order global superconvergence for .
Theorem 5.1. Assume that and . Then, for and , it holds that
Proof. First, we construct an adjoint problem of (1.1).
By taking , (5.2) and (5.3) can then be reduced to the weak form of (1.1) and thus we have the regularity estimate (cf. )
Let in (5.2), it follows from (2.8) and (3.12) that
After integrating in , we have Here the fact that was used.
Now we estimate the right-hand side of (5.6) term by term.
First of all, by Hölder's inequalities, (3.3), and the Sobolev embedding inequality, we obtain that where
Secondly, it follows from (2.9), (2.10) and Hölder's inequalities that
Finally, by Hölder's inequalities, Sobolev embedding inequalities and Lemma 3.2, we have
Therefore, (5.1) holds by combining all estimates together with (5.4) and (5.6).
We finally establish the almost two order global superconvergence in . We define a function and its finite element approximation satisfy that where is the discrete Delta function which satisfies Then the following estimate holds (cf. ): where is the weight function defined by
Furthermore, we have the following estimate.
Lemma 5.2. For and its conjugate index , we have
Proof. Using the Hölder inequality, it is easy to see that Note that (cf. ) the proof is then completed by the norm equivalence in .
The following theorem gives the superconvergence of in .
Theorem 5.3. Assume that and Then, for , with and large enough.
Proof. (5.13) and (3.5) yield that
Then by integrating in , it follows from (5.15) and (5.14) that On the one hand, (2.10) and Lemma 5.2 imply that, for , On the other hand, it follows from (2.7), the Sobolev embedding inequality, (3.3), and (5.16) that Therefore, (5.21) follows from (5.23), (5.24), and (5.25).
6. An Application
In this section, we apply the interpolation postprocessing technique to improve the accuracy of the approximate solution . Let be a quasi-uniform rectangular partition of and let be the space of continuous piecewise polynomial: where
In addition, we assume that in (1.1). By replacing the approximate solution by its interpolation , we can then establish the two order and the almost two order global superconvergence of in and for .
Proof. From (6.4), we have
which together with the triangular inequality and (6.5) yields that
Moreover, for , the following estimates hold (cf. [7, 8]): where
Hence the proof is completed by (6.3) and the estimates for in Theorems 4.1–4.4, 5.1, and 5.3.
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Copyright © 2009 Kening Wang and Shuang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.