Abstract and Applied Analysis

Volume 2012 (2012), Article ID 863125, 19 pages

doi:10.1155/2012/863125

Research Article

Constrained Finite Element Methods for Biharmonic Problem

Rong An and Xuehai Huang

College of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China

Received 12 September 2012; Revised 29 November 2012; Accepted 29 November 2012

Academic Editor: Allan Peterson

Copyright © 2012 Rong An and Xuehai Huang. 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.

Abstract.

This paper presents some constrained finite element approximation methods for the biharmonic problem, which include the symmetric interior penalty method, the nonsymmetric interior penalty method, and the nonsymmetric superpenalty method. In the finite element spaces, the continuity across the interelement boundaries is obtained weakly by the constrained condition. For the symmetric interior penalty method, the optimal error estimates in the broken norm and in the norm are derived. However, for the nonsymmetric interior penalty method, the error estimate in the broken norm is optimal and the error estimate in the norm is suboptimal because of the lack of adjoint consistency. To obtain the optimal error estimate, the nonsymmetric superpenalty method is introduced and the optimal error estimate is derived.

1. Introduction

The discontinuous Galerkin methods (DGMs) have become a popular method to deal with the partial differential equations, especially for nonlinear hyperbolic problem, which exists the discontinuous solution even when the data is well smooth, and the convection-dominated diffusion problem, and the advection-diffusion problem. For the second-order elliptic problem, according to the different numerical fluxes, there exist different discontinuous Galerkin methods, such as the interior penalty method (IP), the nonsymmetric interior penalty method (NIPG), and local discontinuous Galerkin method (LDG). A unified analysis of discontinuous Galerkin methods for the second-order elliptic problem is studied by Arnold et al. in [1].

The DGM for the fourth-order elliptic problem can be traced back to 1970s. Baker in [2] used the IP method to study the biharmonic problem and obtained the optimal error estimates. Moreover, for IP method, the and continuity can be achieved weakly by the interior penalty. Recently, using IP method and NIPG method, Süli and Mozolevski in [3–5] studied the -version DGM for the biharmonic problem, where the error estimates are optimal with respect to the mesh size and are suboptimal with respect to the degree of the piecewise polynomial approximation . However, we observe that the bilinear forms and the norms corresponding to the IP method in [3–5] are much complicated. A method to simplify the bilinear forms and the norms is using interior penalty method. interior penalty method for the biharmonic problem was introduced by Babuka and Zlámal in [6], where they used the nonconforming element and considered the inconsistent formulation and obtained the suboptimal error estimate. Motivated by the Engel and his collaborators’ work [7], Brenner and Sung in [8] studied the interior penalty method for fourth-order problem on polygonal domains. They used the finite element solution to approximate solution by a postprocessing procedure, and the continuity can be achieved weakly by the penalty on the jump of the normal derivatives on the interelement boundaries.

In this paper, thanks to Rivière et al.’s idea in [9], we will study some constrained finite element approximation methods for the biharmonic problem. The continuity can be weakly achieved by a constrained condition that integrating the jump of the normal derivatives over the inter-element boundaries vanish. Under this constrained condition, we discuss three finite element methods which include the symmetric interior penalty method based on the symmetric bilinear form, the nonsymmetric interior penalty method, and nonsymmetric superpenalty method based on the nonsymmetric bilinear forms. First, we study the symmetric interior penalty method and obtain the optimal error estimates in the broken norm and in norm. However, for the nonsymmetric interior penalty method, the norm is suboptimal because of the lack of adjoint consistency. Finally, in order to improve the order of the error estimate, we give the nonsymmetric superpenalty method and show the optimal error estimates.

2. Finite Element Approximation

Let be a bounded and convex domain with boundary . Consider the following biharmonic problem:

where denotes the unit external normal vector to . We assume that is sufficiently smooth such that the problem (2.1) admits a unique solution .

Let be a family of nondegenerate triangular partition of into triangles. The corresponding ordered triangles are denoted by . Let , and . The nondegenerate requirement is that there exists such that contains a ball of radius in its interior. Conventionally, the boundary of is denoted by . We denote

Assume that the partition is quasiuniform; that is, there exists a positive constant such that

Let and be the set of interior edges and boundary edges of , respectively. Let . Denote by the restriction of to . Let with . Then we denote the jump and the average of on by

If , we denote and of on by

Define by

with broken norm

where is the standard Sobolev norm in . Define the broken norm by

where is the seminorm in .

For every and any , we apply the integration by parts formula to obtain

Summing all , we have

Since and , then on . Thus, the previous identity can be simplified as follows:

Now, we introduce the following two bilinear forms:

It is clear that is a symmetric bilinear form and is a nonsymmetric bilinear form. In terms of (2.11) and

the solution to problem (2.1) satisfies the following variational problems:

Let denote the space of the polynomials on of degree at most . Define the following constrained finite element space:

from which we note that the continuity of can be weakly achieved by the constrained condition for all . Next, we define the degrees of freedom for this finite element space. To this end, for any , denote by the three vertices of . Recall that the degrees of freedom of Lagrange element on are , for all with (cf. [10])

Then we modify the degrees of freedom of Lagrange element to suit the constraint of normal derivatives over the edges in . Specifically speaking, the degrees of freedom of are given by

Based on the symmetric bilinear form , the symmetric interior penalty finite element approximation of (2.1) is

Based on the nonsymmetric bilinear form , the nonsymmetric interior penalty finite element approximation of (2.1) is

Moreover, the following orthogonal equations hold:

In order to introduce a global interpolation operator, we first define for and according to the degrees of freedom of by

Due to standard scaling argument and Sobolev embedding theorem (cf. [10]), we have that for every with , ,

where and is independent of . We also suppose that the following inverse inequalities hold:

where is independent of . Then for every , we define the global interpolation operator by . Moreover, from (2.22) there holds

where is independent of .

The following lemma is useful to establish the existence and uniqueness of the finite element approximation solution.

Lemma 2.1. There exists some constant independent of such that

Proof. Introduce a conforming finite element space thanks to Guzmán and Neilan [11]. The advantage of is that the degrees of freedom depend only on the values of functions and their first derivatives. Denote by the interpolation operator from to . Then there holds

where is independent of . Thus, we have

which completes the proof of (2.25) with .

3. Symmetric Interior Penalty Method

In this section, we will show the optimal error estimates in the broken norm and in the norm between the solution to problem (2.1) and the solution to the problem (2.17). First, concerning the symmetric , we have the following coercive property in .

Lemma 3.1. For sufficiently large , there exists some constant such that

Proof. According to the definition of , we have

Using the Hölder’s inequality and the Young’s inequality, we have

where is independent of and is a sufficiently small constant. Thus

Taking , then, for sufficiently large such that , using (2.25) we have

with .

A direct result of Lemma 3.1 is that the discretized problem (2.17) admits a unique solution for sufficiently large .

Lemma 3.2. For all , there holds

where and is independent of .

Proof. For all and , we have

where is independent of . Note that for . Thus, for some constant depending on there holds

That is

From [9], we have

where is independent of . Thus

Substituting (3.11) into (3.7) and using (2.22)-(2.23) give

Theorem 3.3. Suppose that and are the solutions to problems and (2.17), respectively; then the following optimal broken error estimate holds:

where and is independent of .

Proof. According to Lemma 3.1, we have

where we use the orthogonal equation (2.19) and Lemma 3.2. The previous estimate implies

Finally, the triangular inequality and (2.24) yield

Next, we will show the optimal error estimate in terms of the duality technique. Suppose and consider the following biharmonic problem:

Suppose that problem (3.17) admits a unique solution such that

where denotes the norm in and denotes the norm in and is independent of .

Denote by the continuous interpolate operator from to , and satisfies the approximation property (2.22). Then for the solution to problem (3.17), there hold

where is independent of .

Theorem 3.4. Suppose that and are the solutions to problems and (2.17), respectively; then the following optimal error estimate holds:

where and is independent of .

Proof. Setting in (3.17), multiplying (3.17) by , and integrating over , we have

where we use the orthogonal equation (2.19). We estimate two terms in the right-hand side of (3.21) as follows:

where we use the estimate (3.15). In terms of the inequalities (2.22)-(2.23), we have

Substituting the estimates (3.22)-(3.23) into (3.21) yields

4. Nonsymmetric Interior Penalty Method

In this section, we will show the error estimates in the broken norm and in norm between the solution to problem (2.1) and the solution to the problem (2.18). The optimal broken error estimate is derived. However, the error estimate is suboptimal because of the lack of adjoint consistency. According to Lemma 2.1, we have

Moreover, for the nonsymmetric bilinear form , proceeding as in the proof of Lemma 3.2, we have the following lemma.

Lemma 4.1. For all , there holds

where and is independent of .

Theorem 4.2. Suppose that and are the solutions to problems and (2.18), respectively; then there holds

where and is independent of .

Proof. According to (2.20), (4.1), and Lemma 4.1, we have

where is independent of . That is

Using the triangular inequality yields

Theorem 4.3. Suppose that and are the solutions to problems and (2.18), respectively; then there holds

where and is independent of .

Proof. Setting in (3.17), multiplying (3.17) by , and integrating over , we have

where we use . We estimate as follows:

We estimate as follows:

We estimate as follows:

where is some positive constant. Substituting the following estimate

into (4.11) gives

Finally, substituting (4.9)–(4.13) into (4.8), we obtain

5. Superpenalty Nonsymmetric Method

In order to obtain the optimal error estimate for the nonsymmetric method, in this section we will consider the superpenalty nonsymmetric method. First, we introduce a new nonsymmetric bilinear form:

The broken norm is modified to

From Lemma 2.1, it is easy to show that there exists some constant such that

In fact, we have

for . Since the solution to problem (2.1) belongs to , then it satisfies

In this case, the the superpenalty nonsymmetric finite element approximation of (2.1) is

Then, we have the following orthogonal equation:

Let be the continuous interpolated operator defined in Section 3. Observe that on every , and proceeding as in the proof of Lemmas 3.2 and 4.1, we have the following.

Lemma 5.1. For all , there holds

where and is independent of .

Using Lemma 5.1, the following optimal broken error estimate holds.

Theorem 5.2. Suppose that and are the solutions to problems and (5.5), respectively; then there holds

where and is independent of .

The main result in this section is the following optimal error estimate.

Theorem 5.3. Suppose that and are the solutions to problems and (5.5), respectively; then there holds

where and is independent of .

Proof. Let . Multiplying (3.17) by and integrating over , we have

where we use . Proceeding as in the proof of Theorem 4.2, we can estimate and as follows:

The different estimate compared to Theorem 4.3 is the estimate of . Under the new norm , we have

Substituting (5.11)–(5.13) into (5.10) yields the following optimal error estimate:

Acknowledgments

The authors would like to thank the anonymous reviewers for their careful reviews and comments to improve this paper. This material is based upon work funded by the National Natural Science Foundation of China under Grants no. 10901122, no. 11001205, and no. 11126226 and by Zhejiang Provincial Natural Science Foundation of China under Grants no. LY12A01015 and no. Y6110240.

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