Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 829820, 9 pages

http://dx.doi.org/10.1155/2013/829820

Research Article

Superconvergence of a New Nonconforming Mixed Finite Element Scheme for Elliptic Problem

Lifang Pei1,2 and Dongyang Shi²

¹Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471003, China

²School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China

Received 3 January 2013; Revised 30 June 2013; Accepted 5 July 2013

Academic Editor: Trung Nguyen Thoi

Copyright © 2013 Lifang Pei and Dongyang Shi. 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 new nonconforming mixed finite element scheme for the second-order elliptic problem is proposed based on a new mixed variational form. It has the lowest degrees of freedom on rectangular meshes. The superclose property is proven by employing integral identity technique. Then global superconvergence result is derived through interpolation postprocessing operators. At last, some numerical experiments are carried out to verify the theoretical analysis.

1. Introduction

Mixed finite element method (MFEM) is an important branch of FEMs and has been used widely in numerical computation of practical problems. A lot of studies on this aspect have been devoted to the second-order elliptic problems [1–5], in which two approximation spaces of MFEM should satisfy the famous B-B condition [6]. However, since the variable and flux belong to and , respectively, it is not easy to construct a stable MFE space pair. In order to circumvent or ameliorate this deficiency, many approaches have been proposed, such as the least squares FEM [7], stabilization FEM [8], and -Galerkin FEM [9]. Recently, [10] presented a new MFEM, in which B-B condition is automatically satisfied when and meet a relation of inclusion; that is, , where and are finite element approximation spaces of and flux , respectively. This advantage makes the construction of stable MFE space pair extremely simple and convenient. A family of triangular and rectangular conforming MFE space pairs with lower degrees of freedom is constructed in [10], in which the total degrees of freedom of first-order and second-order MFE schemes are about and , respectively; herein denotes the number of nodal points in subdivision. Later, [11] derived the similar results with [10] for conforming MFEM and gave a numerical example.

In this paper, motivated by the idea of [10, 11], we first construct a new nonconforming MFEM (NMFEM). The original variable is approximated by the constrained element space [12] and the flux by piecewise constant vectors space, respectively. Note that the total degrees of freedom of the NMFEM are only about , the lowest on rectangular meshes. We prove that it satisfies B-B condition. Then by the use of integral identity technique, we derive the superclose property for in energy norm and flux in norm. Furthermore, the global superconvergence result with order is obtained through interpolation postprocessing operators. Finally, some numerical results are provided to verify the theoretical analysis. It is observed that, compared with FEM using 4-node quadrilateral (FEM-Q4) and MFEM ( and flux are approximated by piecewise constants and the Raviart-Thomas element, resp.), NMFEM behaves well and has higher rates than MFEM, and it has almost the same rate as FEM-Q4 for in norm, but a higher rate than FEM-Q4 for in energy norm. Moreover, NMFEM is effective and accurate for the diffusion problem studied in [13].

The remainder of this paper is organized as follows. In Section 2, we introduce NMFEM and derive the superclose property and superconvergence results. In Section 3, we carry out some numerical experiments to show the performance of NMFEM.

We will use standard notations for the Sobolev spaces with norm and seminorm , with norm and seminorm , where is an integer. Besides, let and be the norm and norm, respectively. Throughout the paper, denotes a positive constant independent of the mesh parameter and may be different at each appearance.

2. Superconvergence Analysis for NMFEM

Consider the following elliptic problem:

where is a bounded convex polygon domain, .

Let , and then problem (1) is equivalent to the following equations:

We adopt a new mixed variational form in [10] of problem (2). Find such that

where , , , , .

Obviously, and are continuous bilinear functionals, is a continuous linear functional, and for all . Moreover, for , we have . So there exists a constant such that

that is, the B-B condition is satisfied, and therefore (3) has a unique solution .

Let , and then (3) can be written as

Let be a rectangular partition of the domain . For a given element , its four vertices are denoted by , , , , and four edges by , . Let be the reference element with nodes , , , and edges .

Define the affine mapping by

Let be the space of polynomials with degrees defined on , and then the constrained element space is defined by [12]:

where denotes the jump value of across the boundary , and if .

Let denote the number of interior nodes. It has been proven in [12] that and , where are defined associated with nodes of as

We choose the following FE spaces and to approximate and , respectively:

where is the space of constants on . Obviously, the total degree of freedoms of the nonconforming MFE space pair is , and is a norm over .

Then the MFE approximation of problem (3) is to find such that

where .

For , from the affine mapping and the definition of , we can get , and then , that is, . Thus

that is, the discrete B-B condition holds, and (10) has a unique solution .

For all , let , and then we can prove the following two important lemmas.

Lemma 1. For all , the following inequality holds:

Proof. In order to prove (12), we will construct a pair satisfying

Obviously,

On the other hand, for a given arbitrary but fixed , by the discrete B-B condition, there exists a such that

So, for , there holds

As a result, setting , we have

which implies that

where .

Note that

and we have

which follows the desired result (12).

Let and denote the associated interpolation operators of and conforming bilinear element space, respectively. Let be the element interpolation operator [14]; that is, for all , satisfying , where ,. It has been shown in [12] that, for all , because is a bubble function for .

Lemma 2. Assume that , , we have

where is a interpolation operator satisfying .

Proof. Since (23) has been proven by one of the authors in [15], we only need to prove (21) and (22).

In fact, because and are constants on each , we have

which is (21).

On the other hand, note that

Since is a constant vector on and , it follows from integration by parts that

Furthermore, let . Note that and . Employing integral identity technique [16], we have

Similarly,

Therefore, from (25)–(28), we have

The proof is completed.

Now we start to state the following superclose property.

Theorem 3. Assume that and are the solutions of (3) and (10), respectively; , , there holds

Proof. For , from (2) and (10), we have

Applying (31) yields

Using (12) in Lemma 1 and (32), we can obtain

Hence the desired result follows from the interpolation theorem and Lemma 2.

In order to derive global superconvergence for and flux , we introduce the following postprocessing operators and as , , and , , where are the value of on the nodes and are nodes of on macroelement , while consists of the four small elements in , and and are bilinear and biquadratic piecewise polynomials spaces, respectively. It can be checked that the following properties hold:

Then we can have the following superconvergence result.

Theorem 4. Under the assumptions in Theorem 3, there holds

Proof. It follows from (30) in Theorem 3, (34), and the triangle inequality that

which is the desired result.

3. Numerical Experiments

In this section, some numerical examples and comparison with other methods are presented to confirm theoretical analysis and good performance of NMFEM.

We consider the problem (1) with and the exact solution is , and then the flux field can be expressed as . We divide the domain into a family of quasiuniform rectangles with number of . The figures of exact solution of problem (1) and finite element approximation with are plotted in Figures 1, 2, and 3, respectively.

(a)

(b)

Figure 1: The exact flux (a) Numerical approximation (b).

(a)

(b)

Figure 2: The exact flux (a) and numerical approximation (b).

(a)

(b)

Figure 3: The exact displacement (a) and numerical approximation (b).

In Tables 1 and 2, we present the superclose and superconvergence results of the original variable in energy norm and flux in norm with , respectively. It is clearly that , , , and are converged at order 2 with respect to , which coincide with our theoretical analysis in Theorems 3 and 4. In order to describe the results more intuitively, we plot the errors in the logarithm scales in Figure 4.

Table 1: Superclose results of and .

Rate

Rate

0.00210961611276
/
0.00313196226739
/

1.99146324331851

1.99485875617198

1.99788150355682

1.99871061529750

1.99947134660916

1.99967742778650

1.99986789716594

1.99991934319524

1.99996697808428

1.99997983495344

Table 2: Superconvergence results of and .

Rate

Rate

0.00438686071544
/
0.00401808068890
/

0.00109136640979
2.00705335488651

2.03092911988780

2.00193562594041

2.00754934612234

2.00049744480245

2.00186914715453

2.00012537844145

2.00046574790862

2.00003141881610

2.00011631625361

(a)

(b)

Figure 4: The superclose results (a) and the superconvergence results (b).

Moreover, we compare the results of NMFEM with those of FEM using 4-node quadrilateral (FEM-Q4) and MFEM ( and flux are approximated by piecewise constants and the Raviart-Thomas element, resp.).

The convergence rates of errors of in and energy norm are shown in Figures 5 and 6. The comparison of the flux in norm of NMFEM with MFEM is also given in Figure 7. It is observed that (a) the convergence rates of and flux in norm of NMFEM are better than those of MFEM; (b) NMFEM has almost the same rate as FEM-Q4 for in norm, but a higher rate than FEM-Q4 for in energy norm.

Figure 5: Convergence rates of in norm.

Figure 6: Convergence rates of in energy norm.

Figure 7: Convergence rates of the flux in norm.

Furthermore, a numerical experiment is carried out to demonstrate the effectiveness and accuracy of NMFEM for the following diffusion problem:

where is the source term and , is the boundary data, the permeability is a symmetric tensor-valued function such that (a) is piecewise Lipschitz-continuous on and (b) the set of the eigenvalues of is included in (with ) for all .

As [13], we consider the problem (37) with ,

and the analytical solution

Let , we can get the superclose and superconvergence results of in energy norm and flux in norm. The errors are listed in Tables 3 and 4 and plotted in the logarithm scales in Figure 8, respectively.

Table 3: Superclose results of and .

Rate

Rate

0.02493802431889
/
0.03360261961598
/

0.00619633909611
2.00885917091448
0.00820721453088
2.03360913804615

0.00154620480519
2.00268467156460
0.00203799536238
2.00974189629474

2.00070284735607

2.00252791412788

2.00017773032582

2.00063790775102

2.00004455935162

2.00015984995534

Table 4: Superconvergence results of and .

Rate

Rate

0.07704230005041
/
0.06805851358971
/

0.01905713318621
2.01531966736879
0.01670893020781
2.02615627537278

0.00474917374465
2.00458266513090
0.00415565843089
2.00747038382646

0.00118603209709
2.00153348351974
0.00103721683378
2.00235955252584

2.00044337559203

2.00066626183030

2.00011869741141

2.00017656612731

(a)

(b)

Figure 8: The superclose results (a) and the superconvergence results (b).

Acknowledgments

The authors would like to express their sincere thanks to the anonymous referee for his many helpful suggestions, which contribute significantly to the improvement of the paper. The research is supported by the NSF of China (no. 10971203; no. 11271340), Research Fund for the Doctoral Program of Higher Education of China (no. 20094101110006), and Foundation of He’nan Educational Committee (no. 13B110144).

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