A Priori and a Posteriori Error Estimates of a WOPSIP DG Method for the Heat Equation

Zeng, Yuping; Wen, Kunwen; Liang, Fen; Zhu, Huijian

doi:https://doi.org/10.1155/2020/7525676

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 7525676 | https://doi.org/10.1155/2020/7525676

A Priori and a Posteriori Error Estimates of a WOPSIP DG Method for the Heat Equation

Yuping Zeng,¹Kunwen Wen,¹Fen Liang,¹and Huijian Zhu²

Academic Editor: M. I. Herreros

Received26 Aug 2019

Revised26 Dec 2019

Accepted02 Jan 2020

Published17 Apr 2020

Abstract

We introduce and analyze a weakly overpenalized symmetric interior penalty method for solving the heat equation. We first provide optimal a priori error estimates in the energy norm for the fully discrete scheme with backward Euler time-stepping. In addition, we apply elliptic reconstruction techniques to derive a posteriori error estimators, which can be used to design adaptive algorithms. Finally, we present two numerical experiments to validate our theoretical analysis.

1. Introduction

Let be a bounded polygonal domain with Lipschitz boundary , then we consider the following heat equation:where is finite time, is the source term, and stands for the initial data. There has been much research on the a priori and a posteriori error estimates of finite element methods (FEM) for parabolic equations (see [1–12] and the references therein). However, most of the literature on this subject considers only conforming (or nonconforming) FEM in space. The error estimate of discontinuous Galerkin (DG) methods for such problems is still very rare. In the context of DG methods for space variable, see [13–15] for a priori error analysis, and see [16–21] for a posteriori error analysis. Recently, Ern et al. [22, 23] have developed a posteriori error estimates for the parabolic problem with DG discretization in time (see [24] for a posteriori error estimates of nonconforming Crouzeix–Raviart FEM for the heat equation). The object of the present work is to investigate a weakly overpenalized symmetric interior penalty (WOPSIP) method in space for problem (1), combined with an implicit Euler scheme in time.

The WOPSIP method is a kind of nonconsistent discontinuous Galerkin (DG) scheme, which was initially proposed in [25] for solving the second order elliptic equation, therein a priori error estimates were obtained. In recent years, DG methods have received significant attention since they are suitable for -adaptive computations. Besides this, they can also deal with nonhomogeneous boundary conditions and curved boundaries easily and allow for meshes with hanging nodes. Compared with the well-known DG methods in [26], the WOPSIP method has some advantages. For example, it has less computational complexity and thus is easy to implement [25]. Additionally, a high intrinsic parallelism property of the WOPSIP method was investigated in [27]. Therefore, the WOPSIP methods have been further developed to solve biharmonic problems [28] and Stokes equations [29], Reissner–Mindlin plate equations [30, 31], non-self-adjoint and indefinite problems [32], and variational inequalities [33]. In the present work, we shall extend the results of elliptic equations in [25] to the parabolic case. More precisely, the space variable is approximated by the WOPSIP method, and time variable is discretized by the backward Euler scheme. We shall give a detailed a priori and a posteriori error estimates. In this case, one may come across a difficulty that stems from the nonconsistency of the numerical method (for more details, see (34) in Theorem 2). On the contrary, a posteriori error analysis for parabolic problems is more involved than that for elliptic equations, since they involve both spatial error and temporal error. According to a framework stated in [18], we derive a posteriori error estimates which rely on the available estimates for elliptic problems [32, 34].

The rest of our paper is organized as follows. In Section 2, we state some notations and the numerical scheme. We establish a priori error estimates in the energy norm in Section 3. Section 4 is devoted to a posteriori error analysis which is based on the work [18]. We make some conclusions in Section 5. Finally, we provide some numerical results to validate theoretical analysis of a priori error estimates.

2. Preliminaries and WOPSIP Method

Let us first give some notations. For a bounded subdomain , we denote by the standard Sobolev space, associated with norm and seminorm . When , is the standard Lebesgue space , with the inner product defined by . When , we will omit the index .

To deal with functions of time and space, we also introduce the standard Bochner space , which consists of all measurable functions with normfor .

The weak formulation of (1) reads: find with such thatwith .

Let be a family of conforming shape-regular meshes which decompose into triangle elements . Set and . Denote by the set of all edges. Furthermore, , where is the set of interior edges and is the set of edges on . In what follows, stands for the length of the edge e. Given , we let be the space of polynomials of degree at most r on . Moreover, we associate a fixed unit normal with each edge such that for edges on the boundary , is the exterior unit normal vector.

Let e be an interior edge in shared by elements and . For , set , we define the following quantities:

If , set and . Furthermore, for , we also define

Consider the discontinuous finite element space:

The bilinear form of the WOPSIP DG method is defined by (see [25])where stands for the mean of over e, that is,

For the time discretization, we introduce the uniform partition of , with time step and . The backward Euler WOPSIP DG method for solving heat equation (1) is to find for such thatwhere is a projection of onto which will be specified later in Sections 3 and 4.

3. Stability and A Priori Error Analysis

We begin by defining the mesh-dependent norm on as

For the subsequent analysis, we need the following Poincaré–Friedrichs inequality (see [13, 35]):

Here and hereafter, we use C to denote a positive constant which is independent of h and τ, but may have different values at different places.

Moreover, from Lemma 3.1 in [25] we have

Combining (11) and (12) gives

We first prove that the numerical solutions satisfy the following stability result.

Theorem 1. Let be the solution of (9). It holds that, for all ,

Proof. Testing in (9) and using the definition of , we haveThen, using the equation and Cauchy–Schwarz inequality giveswhich together with the inequality (13) and Young’s inequality implies thatThus,Multiplying (18) by and summing from to , we arrive atNoting that , we getThe conclusion (14) follows immediately.

To carry out a detailed error analysis, we introduce which denotes the continuous linear interpolation of u. We then have the following standard estimates (see [36]):

In addition, we introduce the following trace inequality:with e being an edge of K.

Also, we need the elliptic projection operator defined by

It is well known that the projection operator satisfies the following estimates (see [11]).

Lemma 1. For any , it holds that

For convenience, we use the following notation for any function : at each time step . In addition, Taylor expansion yields

Now, we are in a position to state a priori error estimates, which is the main result of this section.

Theorem 2. Let u and be the solutions of (1) and (9), respectively. Assume that , , and satisfying

Then, for all it holds that

Proof. Integrating by parts and using the Taylor formulation (25), we havewhereWe then split the error intoThus, subtracting (9) from (28) and using the definition of and givesTesting in (31), using the definition of and the formula , we obtainWe now bound the first term of the above equation. We then employ the definition of to find thatFurthermore, applying Cauchy–Schwarz inequality and Young’s inequality and using (11), (21), and (22), we can estimate the terms and as follows:Next, we give bounds for . Applying Cauchy–Schwarz inequality, the inequality (13) and Young’s inequality yieldsSimilarly, it holds thatCombining (32)–(36), we obtainMoreover, applying Cauchy–Schwarz inequality for givesOn the contrary,Thus,Plugging (38) and (40) into (37) and then multiplying the result inequality by and summing from to , we obtainNoting that and , and using the estimate (24), we infer thatMoreover, it follows from (24) and (26) thatPlugging (43) into (42) and then combining (30), the triangle inequality yields the desired estimate in (27).

Remark 1. For simplicity, in the present work, we only consider the lowest linear finite element, and the extension of the results to high order methods [37] can be derived straightforwardly.

Remark 2. In this work, we only consider the constant coefficient parabolic equation, and the extension to more practical problems such as equations with variable coefficient can be derived by using the techniques developed in Section 6 in [25]. Additionally, the present work only addresses the conforming mesh, and the extension to meshes with hanging nodes can be also obtained by utilizing approaches stated in Section 6 in [25].

4. A Posteriori Error Analysis

We begin by recalling an a posteriori error estimator for the stationary problem (see [32]), which is a key step in the subsequent error analysis.

Theorem 3. Let be the solution ofand let be the solution satisfying

Then, we have the following a posteriori error estimates:with

Remark 3. It is worth mentioning that, the authors in [34] have proposed a different error estimator, which can also be applied directly in the forthcoming analysis.
The following useful fact is a standard result in studying a posteriori error estimates of DG methods. It shows that, given any discontinuous function , there exists a continuous polynomial function to approximate it (see [34, 38]).

Lemma 2. For any , there exits an decomposition such thatwhere and .

To proceed, we reformulate the DG method (9) to make it be suitable for adaptive computations. At each time step , we assume that a mesh which can be obtained from by locally refining and coarsening . Analogous to Section 3, we denote by the set of edge of , and use to stand for the finite element space with respect to . Let be a general data transfer operator, set , the backward Euler WOPSIP DG method for approximating (1) can be reformulated as: find such thatwhereand is the -projection operator onto . The corresponding mesh-dependent norm is defined by . In view of , we define a function which is piecewise linear continuous in time:for , whereare the Lagrange basic functions.

As in Lemma 2, at each time step , we can decompose intowhere refers to the coarsest common refinement. More precisely, it is the coarsest triangulation which satisfies and . Here, we write to mean that is a refinement of . In addition, we define and .

Moreover, we introduce the discrete elliptic operator bywith . Additionally, we define some elliptic reconstructions which are used commonly in the a posterior error analysis for parabolic equations [6].

Definition 1. The elliptic reconstruction of is defined as the solution of the elliptic problem:Similarly, satisfies

Remark 4. We should point that the DG solution of , denoted by , is also the DG solution . Indeed, we have , for all ; thus, .
Similar to stated in (51), we define byNow, the error e can be decomposed intowhere φ is called the elliptic error and χ is the parabolic error. Furthermore, we define and . Thus, and .
Then, we can obtain the following result, which is a key step to prove the main result in Theorem 4. We remark that similar result can be found in Lemma 5.3 in [39].

Lemma 3. For each , we havefor all .

Proof. Since , this combined with (49), (56), and (57) yieldsfor all . Here, stands for the -projection operator onto . Thus,where in the last line we have used (61).
In addition, for each , it follows from (56)–(58) thatOn the contrary, we haveWe also haveSubstituting (63)–(65) into (62) gives the desired result.
We then define the error estimators as follows:(1)We define the time-stepping estimator as with satisfying(2)Set the data approximation estimator in time to be(3)The estimator with respect to mesh-change is given by(4)The nonconforming part of parabolic estimator is defined as and the nonconforming part of elliptic estimator is given by(5)The space error estimator is where is stated in (47) in Theorem 3. Similarly, we define(6)We also define which can be understood as the nonconforming parabolic part estimator of higher order.We now state the main result of this section, which is largely based on the work [18]. For the sake of completeness, we sketch a proof.

Theorem 4. Let u and be solutions of (1) and (51), respectively. For each , we have

Here, refers to the elliptic error estimator which is defined asand is parabolic error estimator defined by

Proof. Selecting in (60) and noting that and , we deduce thatIntegrating (78) on implies thatThe first term can be bounded byTo estimate , it follows from (67) thatthis together with implies thatFor the term , we first haveWe then estimate as follows:Similarly, the term can be bounded bySubstituting (84) and (85) into (83) yieldsFor the term , we haveObserving thatwe then haveAdapting similar techniques as in the proof of Theorem 5.7 in [18], we can obtain the desired estimate (75).

5. Numerical Experiments

In this section, we give some numerical tests to validate our theoretical analysis in Theorem 2. The numerical scheme (9) shows that, at each time step , we shall solve the following linear system: Given , find such that

In view of this expression, we then implemented our numerical experiments by MATLAB package.

Example 1. We consider problem (1) with , . We choose the source term f such that the exact solution is given by .
The numerical results for the WOPSIP method are stated in Table 1 and Figure 1. The convergence orders shown in Table 1 (which are also the slopes of the line in Figure 1) are consistent with the theoretical results in Theorem 2.

Example 2. We consider problem (1) with , . The exact solution is , and the source term f can be computed accordingly.
The convergence orders stated in Table 2 (see also the slopes of the line in Figure 2) agree with the theoretical analysis in Theorem 2.

6. Conclusion and Future Work

A weakly overpenalized symmetric interior penalty method is proposed and analyzed for solving the heat equation. Optimal a priori error estimates in the energy norm are established. Moreover, we derive a posteriori error estimators which are key indicators to design adaptive algorithms. We only present some numerical tests to validate the a priori error estimate. A posteriori error estimators stated in Section 4 include elliptic and parabolic terms. In particular, the error estimators with respect to the mesh-change operator , such as and , that need to be carefully investigated in the implementation procedure. Thus, the implementation of adaptive algorithms based on the proposed error estimators will be addressed in the further work.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of Guangdong Province, China (grant no. 2018A030307024), National Natural Science Foundation of China (grant no. 11526097), Key Research Platform and Research Project of Universities in Guangdong Province (grant no. 2018KQNCX244), and High Level Innovation Team Program from Guangxi Higher Education Institutions of China (grant no. [2018] 35).

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Copyright © 2020 Yuping Zeng et al. 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.

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