The Scientific World Journal

Volume 2014 (2014), Article ID 297825, 13 pages

http://dx.doi.org/10.1155/2014/297825

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

^{1}School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China^{2}School 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 [3–7], 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 [12–14], 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, 22–25]. 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 4–6.*

*Using the definition of and , we rewrite , , and as
Since estimates of , , and are known, it is enough to estimate , , and . Using Lemmas 4–6, 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 4–6 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 4–6 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*

*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*

*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 .*

*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 1–6 show that new expanded mixed scheme for 2D Sobolev equation with convection term is efficient.*

*6. Concluding Remarks*

*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
*