Journal of Applied Mathematics

Volume 2013 (2013), Article ID 951692, 10 pages

http://dx.doi.org/10.1155/2013/951692

Research Article

A New Characteristic Nonconforming Mixed Finite Element Scheme for Convection-Dominated Diffusion Problem

Dongyang Shi,¹ Qili Tang,1,2 and Yadong Zhang³

¹Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China

²School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, China

³School of Mathematics and Statistics, Xuchang University, Xuchang 461000, China

Received 14 December 2012; Accepted 23 March 2013

Academic Editor: Junjie Wei

Copyright © 2013 Dongyang Shi 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.

A characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation. The optimal order error estimates for both the original variable and the auxiliary variable with respect to the space are obtained by employing some typical characters of the interpolation operator instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At last, we give some numerical results to confirm the theoretical analysis.

1. Introduction

Consider the following convection-dominated diffusion problem:

where is a bounded polygonal domain in with Lipschitz continuous boundary , . and denote the gradient and the divergence operators, respectively.

Model (1) has been widely used to describe the conduction of heat in fluid, the diffusion of soluble minerals or pollutants in ground water, the incompressible miscible displacement in porous media, and so on. The parameters appearing in (1) satisfy the following assumptions [1, 2]:

denotes, for example, the concentration or saturation of soluble substances; represents Darcy velocity of mixed fluid, and a source term; is sufficiently smooth and there exist constants and , such that

It is well known that convection dominated-diffusion problem (1) often presents serious numerical difficulties. The standard numerical methods, such as finite difference method (FDM), FEM and MFEM, usually produce numerical diffusion along sharp fronts. In order to overcome this fatal defect, Douglas et al. [3] combined the method of characteristics with FE procedures so as to reduce the truncation error, and it allows us to use large time steps without lose of accuracy. Moreover, there have appeared many effective discretization schemes concentrating on the hyperbolic nature of the equation, for example, characteristic FD streamline diffusion method [4, 5], Eulerian-Lagrangian method [6, 7], characteristic-finite volume element method [2, 8, 9], characteristics-mixed covolume method [10, 11], the modified method of characteristic-Galerkin FE procedure [12], characteristic nonconforming FEM [13–15], characteristic MFEM [16–19] and expanded characteristic MFEM [1, 20], and so forth.

As for the characteristic MFEM or expanded characteristic MFEM, the convergence rates of and in existing literature were suboptimal [11, 18, 21, 22] and the convergence analysis was valid only to the case of the lowest order MFE approximation [10, 17]. So far, to our best knowledge there are few studies on the optimal order error estimates except for [23], in which a family of characteristic MFEM with arbitrary degree of Raviart-Thomas-Nédélec space in [24, 25] for transient convection diffusion equations was studied.

Recently, based on the low regularity requirement of the flux variable in practical problems, a new mixed variational form for second elliptic problem was proposed in [26]. It has two typical advantages: the flux space belongs to the square integrable space instead of the traditional , which makes the choices of MFE spaces sufficiently simple and easy; the LBB condition is automatically satisfied when the gradient of approximation space for the original variable is included in approximation space for the flux variable. Motivated by this idea, this paper will construct a characteristic nonconforming MFE scheme for (1) with a new mixed variational formulation. Similar to the expanded characteristic MFEM, the coefficient of (1) in this proposed scheme does not need to be inverted; therefore, it is also suitable for the case when is small. By employing some distinct characters of the interpolation operators on the element instead of the mixed or expanded mixed elliptic projection used in [1, 17, 20] which is an indispensable tool in the traditional characteristic MFEM analysis, the order error estimate in -norm for original variable , which is one order higher than [1, 20] and half order higher than [18], is derived, and the optimal error estimates with order for auxiliary variable in -norm and for in broken -norm are obtained, respectively. It seems that the result for in broken -norm has never been seen in the existing literature by making full use of the high-accuracy estimates of the lowest order Raviart-Thomas element proved by the technique of integral identities in [27] and the special properties of nonconforming element (see Lemma 1 below).

The paper is organized as follows. Section 2 is devoted to the introduction of the nonconforming FE approximation spaces and their corresponding interpolation operators. In Section 3, we will give the construction of the new characteristic nonconforming MFE scheme and two important lemmas, and the existence and uniqueness of the discrete scheme solution will be proved. In Section 4, the convergence analysis and optimal error estimates for both the original variable and the flux variable are obtained. In Section 5, some numerical results are provided to illustrate the effectiveness of our proposed method.

Throughout this paper, denotes a generic positive constant independent of the mesh parameters and with respect to domain and time .

2. Construction of Nonconforming MFEs

As in [28], we frequently employ the space of square integrable functions with scalar product and norm

We also employ the Sobolev space , of functions such that for all , equipped with the norm and seminorm

The space denotes the closure of the set of infinitely differentiable functions with compact supports in . For any Sobolev space ,  is the space of measurable -valued functions of , such that if , or such that if .

We now introduce the nonconforming MFE space described in [29] for and summarize it as follows.

Let be a polygon domain with edges parallel to the coordinate axes on plane, and let be a rectangular subdivision of satisfying the regular condition [30]. For a given element , denote the barycenter of element by , denote the length of edges parallel to -axis and -axis by and , respectively, .

Let be the reference element on plane and four vertices , ,  , and , the four edges , , , and . Then there exists an affine mapping as

Define the FE spaces ,   by

where , , ,  , .

The interpolation operators on are defined as follows:

Then the associated nonconforming element space [29] and lowest order Raviart-Thomas element space [25, 27] are defined as

respectively, where represents the jump value of across the boundary , and if .

Similarly, the interpolation operators and are defined as

3. New Characteristic Nonconforming MFE Scheme and Two Lemmas

Let and be the characteristic direction associated with  , such that

Then (1) can be put in the following system:

By introducing and using Green's formula, we obtain the new characteristic mixed form of (11). Find , such that

Let , , and . When solving , we would like to make the scheme as implicit as possible by using of the characteristic vector . Denote and

similar to [1, 3], and then we have the following approximation:

This leads to the following characteristic nonconforming MFE scheme. Find , , such that

where , . Generally speaking, are not node values and should be derived by interpolation formulas on .

Remark 1. In [1], the expanded characteristic MFE scheme was presented by introducing two new auxiliary variables which avoided the inversion of the coefficient when is small. The new mixed schemes (15a), (15b), and (15c) not only keep the advantage of expanded characteristic MFE scheme, but also donot need to solve three variables.

Now, we prove the existence and uniqueness of the solution of (15a), (15b), and (15c).

Theorem 1. Under assumption (A3), there exists a unique solution to the schemes (15a), (15b), and (15c).

Proof. The linear system generated by (15a), (15b), and (15c) is square, so the existence of the solution is implied by its uniqueness. From (15a), (15b), and (15c), we have

Let and be zero, and thus is zero too; taking in (16) and adding them together, we have

Thus assumption (A3) implies that . The proof is complete.

To get error estimates, we state the following two important lemmas.

Lemma 1 (see [27, 29, 31]). Assume that , for all , and then there hold

where is a norm on , and denotes the outward unit normal vector on .

Lemma 2 (see [1, 3]). Let , and , where function and its gradient are bounded, then

where .

4. Convergence Analysis and Optimal Order Error Estimates

In this section, we aim to analyze the convergence analysis and error estimates of characteristic nonconforming MFEM. In order to do this, let

Taking in (12) yields

From (23a), (23b), (15a), (15b), and (15c) we get

We are now in a position to prove the optimal order error estimates.

Theorem 2. Let and be the solutions of (12), (15a), (15b), and (15c), respectively, and assume that . Then under assumption (A3), we have

Proof. Taking in (24a) and in (24b), and adding them, we have

On the one hand, we consider the right hand of (28).

Using the method similar to [3], we have

can be estimated as

By Lemma 2, we obtain

It follows from Lemma 1 that

Let . By Lemma 1, we have

On the other hand, the left hand of (28) can be bounded by

where the inequality proved in [3] is used in the last step.

Combining (29)–(34) with (28) gives

Taking , multiplying (35) by , summing over from to , and noticing that , we obtain

It follows from discrete Gronwall’s lemma that

From (37) we get the optimal order error estimate of rather than . So we start to reestimate in the following manner and derive the estimation of simultaneously.

Firstly, choosing in (24a) and in (24b), and adding them, we have

The left hand can be estimated as

and can be bounded by

From (38)–(40), we get

Multiplying (41) by and summing over in time from to yield

Secondly, we take and must approach zero in such a way that and satisfy

and by inverse inequality, we have

At the same time, using Lemma 2, we obtain

From (42)–(45), taking suitable small such that , we have

Finally, applying discrete Gronwall’s lemma yields

In order to derive (27), set in (24b) and employ Lemma 1 and assumption (A3) to give

Combining (47) with (48) yields

By using of interpolation theory and the triangle inequality, (37), (47), and (49) lead to (25), (26), and (27), respectively, which are the desired results.

Remark 2. From (37), we have

This byproduct can be regarded as the superclose result between and in mean broken -norm. It seems that both (25) and (50) have never been seen in the existing studies. At the same time, by employing the new characteristic nonconforming MFE scheme, we can also obtain the same error estimate of (27) as traditional characteristic MFEM [10].

Remark 3. From the analysis of Theorem 2 in this paper, we may see that Lemma 1 is the key result leading to the successful optimal order error estimations. If we want to get higher order accuracy, similar to Lemma 1, the nonconforming finite elements for approximating should also possess a very special property, that is, the consistency error estimates with order, and satisfy (18). For the famous nonconforming Wilson element [32] whose shape function is , by a counter-example, it has been proven in [32] that its consistency error estimate is of order and cannot be improved any more. For the rotated bilinear element [33] whose shape function is , although its consistency error with order and on square meshes is satisfied, the second term of (18) is not valid. Thus when they are applied to (1) on new characteristic mixed finite element scheme, up to now, the optimal order error estimates of (25), (26), and (27) cannot be obtained directly.

5. Numerical Example

In order to verify our theoretical analysis in previous sections, we consider the convection-dominated diffusion problem (1) as follows:

with , and .

The right hand term is taken such that   are the exact solutions.

We first divide the domain into and equal intervals along -axis and -axis and the numerical results at different times are listed in Tables 1, 2, and 3 and pictured in Figures 1, 2, 3, and 4, respectively. denotes the characteristic nonconforming MFE solution of the problem (15a), (15b), and (15c). represents the time step and the experiment is done with . stands for the convergence order.

Table 1: Numerical results of .

0.75277
/
0.75017
/
0.66433
/

0.42984
0.81
0.41849
0.84
0.35474
0.91

0.21758
0.99
0.21412
0.97
0.17552
1.02

0.55291
/
0.42211
/
0.40937
/

0.29234
0.92
0.23117
0.87
0.21120
0.96

0.14466
1.02
0.10807
1.10
0.09343
1.18

Table 2: Numerical results of .

0.0298190
/
0.0276370
/
0.0223240
/

0.0073087
2.03
0.0062445
2.15
0.0048038
2.22

0.0020769
1.82
0.0017926
1.80
0.0013309
1.85

0.0198730
/
0.0175900
/
0.0154090
/

0.0044472
2.16
0.0041982
2.07
0.0039150
1.98

0.0011894
1.90
0.0010738
1.97
0.0009466
2.05

Table 3: Numerical results of .

/

/

/

1.05

1.18

1.19

1.03

1.07

1.06

/

/

/

1.20

1.18

1.14

1.05

1.07

1.09

Figure 1: Errors at .

Figure 2: Errors at .

Figure 3: Errors at .

Figure 4: Errors at .

It can be seen from the above Tables 1, 2, and 3 that and are convergent at optimal rate of and is convergent at optimal rate of , respectively, which coincide with our theoretical investigation in Section 4.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (Grant nos. 10971203, 11101384, and 11271340) and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20094101110006). The author would like to thank the referees for their helpful suggestions.

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