Journal of Applied Mathematics

Volume 2013 (2013), Article ID 934973, 9 pages

http://dx.doi.org/10.1155/2013/934973

Research Article

Expanded Mixed Finite Element Method for the Two-Dimensional Sobolev Equation

Qing-li Zhao,1,2 Zong-cheng Li,² and You-zheng Ding²

¹School of Mathematics, Shandong University, Jinan 250100, China

²School of Science, Shandong Jianzhu University, Jinan 250101, China

Received 23 April 2013; Revised 7 June 2013; Accepted 7 June 2013

Academic Editor: Carlos Conca

Copyright © 2013 Qing-li Zhao 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.

Abstract.

Expanded mixed finite element method is introduced to approximate the two-dimensional Sobolev equation. This formulation expands the standard mixed formulation in the sense that three unknown variables are explicitly treated. Existence and uniqueness of the numerical solution are demonstrated. Optimal order error estimates for both the scalar and two vector functions are established.

1. Introduction

In this paper, we consider the following Sobolev equation:

in a bounded domain with Lipschitz continuous boundary . Equation (1) has a wide range of applications in many mathematical and physical problems [1, 2], for example, the percolation theory when the fluid flows through the cracks, the transfer problem of the moisture in the soil, and the heat conduction problem in different materials. So there exists great and actual significance to research Sobolev equation. Up till now, there are some different schemes studied to solve this kind of equation (see [1–4] for instance).

The mixed finite element method, which is a finite element method [5] with constrained conditions, plays an important role in the research of the numerical solution for partial differential equations. Its general theory was proposed by Babuska [6] and Brezzi [7]. Falk and Osborn [8] improved their theory and expanded the adaptability of the mixed finite element method. The mixed finite element method (see [9, 10] for instance) is wildly used for the modeling of fluid flow and transport, as it provides accurate and locally mass conservative velocities.

The main motivation of the expanded mixed finite element method [11–15] is to introduce three (or more) auxiliary variables for practical problems, while the traditional finite element method and mixed finite element method can only approximate one and two variables, respectively. The expanded mixed finite element method also has some other advantages except introducing more variables. It can treat individual boundary conditions. Also, this method is suitable for differential equation with small coefficient (close to zero) which does not need to be inverted. Consequently, this method works for the problems with small diffusion or low permeability terms in fluid problems. Using this method, we can get optimal order error estimates for certain nonlinear problems, while standard mixed formulation sometimes gives only suboptimal error estimates.

The object of this paper is to present the expanded mixed finite element method for the Sobolev equation. We conduct theoretical analysis to study the existence and uniqueness and obtain optimal order error estimates. The rest of this paper is organized as follows. In Section 2, the mixed weak formulation and its mixed element approximation are considered. In Section 3, we prove the existence and uniqueness of approximation form. In Section 4, some lemmas are given. In Section 5, optimal order semidiscrete error estimates are established.

Throughout the paper, we will use , with or without subscript, to denote a generic positive constant which does not depend on the discretization parameter . Vectors will be expressed in boldface. At the same time, we show a useful -Cauchy inequality

2. Mixed Weak Form and Mixed Element Approximation

For and any nonnegative integer, let

denote the Sobolev spaces [16] endowed with the norm

(the subscript will always be omitted).

Let with the norm . The notation will mean or . We denote by the inner product in either or ; that is,

The notation denotes the -inner product on the boundary of

To formulate the weak form, let ; we introduce two vector variables

Letting , we rewrite (1) as the following system:

Let be the unit exterior normal vector to the boundary of . Then (8) is formulated in the following expanded mixed weak form: find , such that

where

Let be a quasiregular polygonalization of (by triangles or rectangles), with being the maximum diameter of the elements of the polygonalization. Let be a conforming mixed element space with index and discretization parameter . is an approximation to . There are many conforming (or compatible) mixed element function spaces such as Raviart-Thomas elements [17], BDFM elements [18, 19]. Some RT type mixed elements are listed in Table 1. Here, is the polynomial up to order in two-dimensional domain used in triangle, while is the polynomial up to in each dimension used in rectangle.

Table 1: Some RT type mixed elements.

Dim
Element

2D
Triangle

2D
Rectangle

Replacing the three variables by their approximation, we get the expanded mixed finite element approximation problem: find , such that

where

The error analysis next makes use of three projection operators. The first operator is the Raviart-Thomas projection (or Brezzi-Douglas-Marini projection) ; satisfies

The following approximation properties are well known.

The other two operators are the standard -projection [5] and onto and , respectively,

They have the approximation properties

The two projections and preserve the commuting property

3. Existence and Uniqueness of Approximation Form

In this section, we consider the existence and uniqueness of the solution of (11).

Lemma 1. Equation (11) has a unique solution.

Proof. In fact, this equation is linear; it suffices to show that the associated homogeneous system

has only the trivial solution. In the first equation of (19), if we take and , respectively, then we have that

By (20), it is easy to see that

Choosing , , and in (19), we get

In the third equation of (19), letting and , respectively, then

Using the -Cauchy inequality to (22), we have

Note that (21), we deserve

From (27), we know

Using Gronwall’s inequality to (28), we can prove . Further, from (26) and (23), we know that , . The proof is completed.

4. Some Lemmas

In the study of parabolic equations, we usually introduce a mixed elliptic projection associated with our equations. Define a map: find , such that

Similarly to Lemma 1, we can prove that system (29) has a unique solution.

Now we give some error estimates of . Define

System (29) can be rewritten as follows:

Now we consider the estimates of and .

Lemma 2. Let be the solution of system (31). If , , and are sufficiently smooth, then there exist positive constants such that

where for , and for .

Proof. Let be the solution of the following problem:

Then we know

For , from the second equation of (31), we have that

It is easy to see that

Now, we estimate ,

We turn to consider ,

where is the piecewise constant interpolation of function . Using the previous estimates, we obtain

So we deserve

Applying Gronwall’s inequality to (40), we have

Noting the estimates of , , and , the proof is completed.

Define

Lemma 3. Let and be the solution of (19) and (29), respectively. If , , and are sufficiently smooth, then there exist positive constants such that

(I)if , then

(II)if , then

(III)if , then

Proof. For , the proof proceeds in three steps as follows.

(I) In the first equation of (29), taking , we know

which implies that

Choosing , note that and , we get

Combing (15) with (56), we can prove (43).

(II) In (29), letting , together with (43), we obtain

Hence

In the third equation of (29), choosing , we get

So we know

Note that (44) and (45), we obtain

which implies that

Now we estimate ,

It is easy to see that

By the properties of projection , we get

which proves (49) and (53).

Now we estimate ,

where

From (66) and (67), we know

By Lemma 2, if , we have that

If , we have that

Using the estimate of , we obtain

If , we get

If , we get

(III) By Lemma 2, we have that

If , we know

If , we know

The proof is completed.

Remark 4. The estimate results of (46)–(53) are optimal. If , the estimate results of (43)–(45) are superconvergent.

5. Main Result

In this section, we consider error estimates for the continuous-in-time mixed finite element approximation. Define

Theorem 5. Let and be the solution of (9) and (11), respectively. If , , and are sufficiently smooth, then there exist positive constants such that

(I)if , then

(II)if and , then

Proof. From (9) and (19), we have the following error equation:

In (85), choosing , , and , we have

In the second equation of (85), letting , we know

Further, using -inequality to (86), we get

Hence,

Using Gronwall’s inequality to (89), we obtain

Further, we know

Noting the first equation of (78), we get

So we have that

Together with the results of Lemmas 2 and 3, the proof is completed.

Remark 6. The estimate results of (78)–(84) are optimal.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (nos. 11171190, 11101246, and 11101247) and the Natural Science Foundation of Shandong Province (ZR2011AQ020). The authors thank the anonymous referees for their constructive comments and suggestions, which led to improvements in the presentation.

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