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The Scientific World Journal
Volume 2014, Article ID 297825, 13 pages
http://dx.doi.org/10.1155/2014/297825
Research Article

A New Expanded Mixed Element Method for Convection-Dominated Sobolev Equation

1School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China
2School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

Received 30 August 2013; Accepted 2 December 2013; Published 18 February 2014

Academic Editors: G. Fernandez-Anaya and L. Guerrini

Copyright © 2014 Jinfeng Wang 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

We propose and analyze a new expanded mixed element method, whose gradient belongs to the simple square integrable space instead of the classical H(div; Ω) space of Chen’s expanded mixed element method. We study the new expanded mixed element method for convection-dominated Sobolev equation, prove the existence and uniqueness for finite element solution, and introduce a new expanded mixed projection. We derive the optimal a priori error estimates in -norm for the scalar unknown and a priori error estimates in -norm for its gradient λ and its flux σ. Moreover, we obtain the optimal a priori error estimates in -norm for the scalar unknown u. Finally, we obtained some numerical results to illustrate efficiency of the new method.

1. Introduction

We consider the following Sobolev equation with convection term: where is a bounded convex polygonal domain in with Lipschitz continuous boundary and is the time interval with . and are given functions, coefficients , are smooth and bounded functions, coefficient is a bounded vector, and for some positive constants , , , , and .

Sobolev equations are a class of important evolution partial differential equations and have a lot of applications in many physical problems, such as the porous theories concerned with percolation into rocks with cracks, the heat conduction problems in different mediums, and the transport problems of humidity in soil. In [1], the finite element method for nonlinear Sobolev equation with nonlinear boundary conditions was studied. In [2], a discontinuous Galerkin method for Sobolev equation was studied. In [37], some mixed finite element methods for Sobolev equations are studied and analyzed.

In 1994, Chen [8, 9] developed and studied an expanded mixed element method and proved some mathematical theories for second-order linear elliptic equation. Compared to standard mixed element methods the expanded mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, its gradient, and its flux. From then on, the expanded mixed element method has been applied to solving other partial differential equations [10]. At the same time, many researchers proposed and studied some new numerical methods based on Chen’s expanded mixed method, such as expanded mixed hybrid methods [11], two-grid expanded mixed finite element method [1214], expanded characteristic-mixed element method [15], expanded mixed covolume method [16, 17], and expanded positive definite mixed method [18].

In 2011, we developed and analyzed a new expanded mixed finite element method [19] for elliptic equations based on the mixed schemes [20, 21] which have been studied for some partial differential equations [4, 2225]. Compared to Chen’s expanded mixed method, the gradient for the new expanded mixed method belongs to the simple square integrable space instead of the classical space. In this paper, we will study the new expanded mixed element method for convection-dominated Sobolev equation. We will give the proof for the existence and uniqueness of the solution for semidiscrete scheme and a new expanded mixed projection and the proof of its uniqueness. We will prove the optimal a priori error estimates in -norm for the scalar unknown and a priori error estimates in -norm for its gradient and its flux . In particular, we obtained the optimal a priori error estimates in -norm for the scalar unknown . Finally, we obtained some numerical results to confirm our theoretical analysis.

Throughout this paper, will denote a generic positive constant which is free of the space-time parameters and . At the same time, we denote the natural inner product in or by with the corresponding norm . The other notations and definitions of Sobolev spaces as in [26] are used.

2. New Expanded Mixed Formulation

Introducing the two auxiliary variables , , we obtain the following first-order system for (1): Using Chen’s expanded mixed method, the mixed weak formulation for problem (1) is to find such that where or , , and .

In this paper, the new expanded mixed weak formulation of (3) is to find such that Then, the semidiscrete mixed finite element scheme for (5) is to determine such that where is chosen as the finite element pair as follows: From [20, 21], we find that satisfies the so-called discrete Ladyzhenskaya-Babuska-Brezzi condition.

Remark 1. Compared to Chen’s expanded mixed weak formulation (4), the gradient in the scheme (5) belongs to the simple square integrable space instead of the classical space. Obviously, the regularity requirements on the solution reduced.

Theorem 2. There exists a unique discrete solution to semidiscrete scheme (6).

Proof. Let and be bases of and , respectively. Let and substituting these expressions into (6) and choosing , , and , the problems (6) can be written as follows: find such that, for all, where It is easy to see that and are invertible matrixes. From (9), the initial value problems can be written as follows:
Substitute (11)(b) into (11)(c) to get
Substituting (11)(b) and (12) into (11)(a), we obtain where is a unit matrix.
Thus, by the theory of differential equations [27], (13) has a unique solution ; then (12) and (11)(b) have unique solutions and , respectively. Equivalently (6) has a unique solution.

3. Error Estimates for Semidiscrete Scheme

In order to analyze the convergence of the method, we first introduce the new expanded mixed elliptic projection associated with our equations.

Let be given by the following mixed relations:

Theorem 3. There exists a unique solution to the new expanded mixed elliptic projection (14).

Proof. Noting that mixed elliptic projection system (14) is linear, it suffices to prove the associated homogeneous system has the trivial solution.
Choose in (15)(a), in (15)(b), and in (15)(c) to obtain Add the three equations to get Integrate (17) with respect to time from to and use Cauchy-Schwarz inequality and Young inequality to obtain Taking in (15) and using Poincaré inequality, we obtain Substitute (19) into (18) and use Gronwall lemma to obtain From (20), we have Combining (19), (21), and (15)(c), we get Using (21) and (22), we get

In the following discussion, we will give some important lemmas based on new mixed scheme.

Lemma 4. There exists a linear operator such that

Lemma 5. For the linear operator of Lemma 4, one has

Lemma 6. There exists a linear operator such that

From [20, 21], we can obtain the proof for Lemmas 46.

Using the definition of and , we rewrite , , and as Since estimates of , , and are known, it is enough to estimate , , and . Using Lemmas 46, we rewrite (14) as We discuss the following approximation properties for system (29).

Lemma 7. There is a constant independent of such that

Proof. Choose in (29)(a), in (29)(b), and in (29)(c) to obtain Add the three equations and use Cauchy-Schwarz inequality to get
Integrate (35) with respect to time from to to obtain Using the Gronwall lemma, we have Differentiating (34)(b) and taking , we obtain
Choose in (34)(a) and in (34)(c) to obtain
Combining (38) and (39), we have
Substitute (37) into (40) to obtain
Choose in (29)(c) and use Cauchy-Schwarz inequality (41) to obtain
Choose in (29)(b) and use (37) and Cauchy-Schwarz inequality to obtain
Combining (37), (42), (43), and Lemmas 46 and using the triangle inequality, we get the conclusion of Lemma 7.

Lemma 8. There is a constant independent of such that

Proof. To estimate terms , , and , we consider the following auxiliary elliptic problem: Use (45) and Lemmas 46 to obtain
From (46), we obtain
Using similar method to , we can obtain
Combining (33) and (47), we obtain
Using (47)–(49), we obtain the conclusion of Lemma 8.

For a priori error estimates, we decompose the errors as

Using (5)-(6) and (14), we can get the error equations

We will prove the error estimates for semidiscrete scheme.

Theorem 9. Assume that and ; then one has the following estimates: where .

Proof. Choose in (51)(a), in (51)(b), and in (51)(c) to obtain Adding the above three equations, and using Cauchy-Schwarz inequality and Young inequality, we have
Integrate with respect to time from to to obtain Using Gronwall lemma, we obtain Take in (51)(b) to have Differentiating (51)(b) and taking , we obtain Choose in (51)(a) and in (51)(c) to obtain Adding (58), (59)(a), and (59)(c) and using Cauchy-Schwarz inequality, Young inequality, and (56), we have So, we have Choosing in (51)(c) and using (56) and (61), we have Combining Lemmas 7 and 8, (56), (57), (61), (62), and the triangle inequality, we obtain the error estimate for Theorem 9.

4. Fully Discrete Scheme and Error Estimates

In this section, we get the error estimates of fully discrete schemes. For the backward Euler procedure, let be a given partition of the time interval with step length and nodes , for some positive integer . For a smooth function on , define and .

Equation (5) has the following equivalent formulation: where Now we formulate a completely discrete procedure. Find , such that For the fully discrete error estimates, we now split the errors

We will prove the theorem for the fully discrete error estimates.

Theorem 10. Assume that and ; then there exists a positive constant independent of and such that

Proof. Using (14), (63), and (65) at , we get the error equations
Choose in (68)(a), in (68)(b), and in (68)(c) to obtain Adding the above three equations, we obtain Multiplying by and summing (70) from to , the resulting equation becomes Choose in such a way that for , , . Then we use Cronwall’s lemma to obtain Note that Substitute (73) to (72) to get Taking in (68)(b), we have From (68)(b), we get Choose in (76), in (68)(a), and in (68)(c) to obtain
Adding the three equations, we obtain
Using (78) and (74), we get
Taking in (68)(c), we get
Substitute (79) into (80) to get
Combining (33)-(24), (74), (75), (81), and the triangle inequality, we complete the proof.

5. Numerical Example

In order to illustrate the efficiency of the new expanded mixed element method, we consider the following initial-boundary value problem of 2D Sobolev equation with the convection term: where , , , , and , and is chosen so that the exact solution for the scalar unknown function is the corresponding exact gradient function is and its exact flux function is

We divide the domain into the triangulations of mesh size uniformly, consider the piecewise linear space with index for the scalar unknown function and the piecewise constant space with index for the gradient and the flux , use the backward Euler procedure with uniform time step length , and obtain some convergence results for , , , and with , , in Table 1. At the same time, we show the exact solutions , , and in Figures 1, 3, and 5, respectively, and the corresponding numerical solutions , and in Figures 2, 4, and 6, respectively, with , and .

tab1
Table 1: Errors and order of convergence.
297825.fig.001
Figure 1: Surface for exact solution .
297825.fig.002
Figure 2: Surface for numerical solution .
fig3
Figure 3: Surface for exact solution .
fig4
Figure 4: Surface for numerical solution .
fig5
Figure 5: Surface for exact solution .
fig6
Figure 6: Surface for numerical solution .

It is easy to see that we obtained the optimal error estimates for in -norm, -norm, and the error estimates for and in -norm, which confirm the theoretical results in this paper, in Table 1. The numerical results in Table 1 and Figures 16 show that new expanded mixed scheme for 2D Sobolev equation with convection term is efficient.

6. Concluding Remarks

In this paper, a new expanded mixed finite element method is proposed and studied for Sobolev equation with convection-term. The proof for the existence and uniqueness of the solution for semidiscrete scheme, the new expanded mixed projection, and the proof of its uniqueness are given. The optimal a priori error estimates in for the scalar unknown and the a priori error estimates in -norm for its gradient and its flux are proved. Especially, the optimal a priori error estimates in -norm for the scalar unknown are derived. Finally, some numerical results are provided to confirm our theoretical analysis.

In the near future, the new expanded mixed method will be applied to other evolution equations such as evolution integrodifferential equations, hyperbolic wave equations, and nonlinear evolution equations. And the new characteristic expanded mixed finite element method for Sobolev equation will be studied. The new expanded characteristic-mixed weak formulation is to find such that where