Abstract

In this paper, we suggest a new patch condition for nonconforming mixed finite elements (MFEs) on parallelepiped and provide a framework for the convergence. Also, we introduce a new family of nonconforming MFE space satisfying the new patch condition. The numerical experiments show that the new MFE shows optimal order convergence in and -norm for various problems with discontinuous coefficient case.

1. Introduction

The finite element method has achieved great success in many fields, and it has become a powerful tool for solving partial differential equations [1–3]. The main idea of the finite element method is using a finite dimensional space to approximate the exact solution on the given space according to a certain kind of variational principle. In the finite element discretizations, a basic distinction can be made between conforming and nonconforming methods. When the finite element space is a subspace of the solution space, the method is called conforming. In this case, the error between the true solution and the finite element method (FEM) solution is bounded by the distance between the FEM space and the given space by Céa’s lemma. Meanwhile, the nonconforming finite element space is not contained in the space the exact solution lives in. Hence, extra error committed by the nonconformity has to be estimated.

There are some situations in which finite element methods for the primary variable do not yield satisfactory results, such as elliptic problem with large jumps in the diffusion coefficient. Sometimes, other quantities such as Darcy velocity along with pressure of the flow in the porous media become variables of main interest. In this case, the mixed finite element method (MFEM) is preferred. Many mixed finite element methods have been developed since it was first suggested in the late 1970s [4–6]. The idea of the mixed methods is to introduce the velocity as a new variable and change the given equation into a system of equations. By discretizing this system, we can compute two variables, velocity and pressure, simultaneously and expect a more accurate velocity. MFEM has been used in many applications such as porous media problem [7, 8] and chemical engineering [9].

The nonconforming approaches have been widely studied for Lagrangian finite elements. So far, all the well-known mixed finite element (MFE) space are conforming in the sense that the space is contained in , defined as the space of all vector functions whose divergence belongs to . It is natural to ask whether there exists a nonconforming counterpart of the MFE space. Hiptmair [10] has investigated some conditions for the nonconforming MFE space. But, under the conditions suggested there, they only show suboptimal convergence. Meanwhile, a family of high-order nonconforming MFE space was introduced in [11] a few years ago, and numerical examples show the optimal order of convergence. However, there is no analysis.

In this paper, we suggest a new condition under which the nonconforming MFEM may have optimal convergence. In addition, we introduce another family of the nonconforming MFE space which satisfies this condition. To the author’s best knowledge, nonconforming element having optimal order has not been suggested by others.

The organization of this paper is as follows: in the next section, we present the model problem. In Section 3, we introduce nonconforming mixed finite element spaces on parallelepiped in together with a new patch condition. A framework for the convergence is given in Section 4. We give numerical experiments in Section 5.

2. Model Problem

Given , a simply connected bounded Lipschitz polyhedral domain with connected boundary , we consider the following second-order elliptic problem:where is assumed to be uniformly positive definite and bounded. And is a given function in . To write the given equation into a mixed system, we use Darcy’s law, . Then, we can rewrite problem (1) in the following mixed form:

Denote by and the usual Sobolev spaces with obvious norms. Then, we have the following variational form for (2): find such that

For the convenience of the presentation, we let

Then, saddle point problems (3) and (4) can be expressed simply as follows: find such thatwhere indicates the inner product in . If familiar inf-sup condition holds, then problem (6) has a unique solution [12].

3. Nonconforming Mixed Finite Element Spaces

The main idea of MFEM is solving problem (6) over suitable locally defined finite dimensional spaces. Their construction depends on triangulations of . Let and be a family of partitions of into parallelepiped obtained by uniform division having each side length .

For the construction of the nonconforming MFE space , we require that

That is, a function in the space is locally in but not in over the triangulation . A lack of continuity of normal components across interelement boundaries gives rise to a nonconforming approximation. But, we still require some local conformity.

For any domain in or , let be the space of polynomials of total degree and or be the space of polynomials of degree less or equal to , respectively, in each variable.

3.1. Patch Conditions

First, we recall a well-known type of “patch condition”: let be the common face of two adjacent elements and .where is an outer unit normal vector to each element. For example, Hiptmair used this condition together with the assumption of continuous interpolation and showed a suboptimal error estimate [10]. However, for the 3D case (parallelepiped), we see that such condition is not enough to guarantee the existence of continuous interpolation which is necessary to derive an approximation. With hypothesis (H1) holding for is not enough to determine a continuous interpolation over the whole domain . In fact, if such an interpolation exists, then (H1) holds for , which would imply the conformity of the space . Thus, to study a nonconforming MFE, we need a stronger patch condition but not strong enough to make the space fully conforming.

So, we suggest new patch conditions (H2): for all vertical faces and all , we have

For horizontal faces and all , we have

This means that the moments up to order of the discrete velocity are continuous across horizontal interelement boundaries with respect to , and the moments across vertical interelement boundaries are continuous up to only.

Now, we introduce a new nonconforming MFE. We denote by the set of all polynomials of except those having the form for .

Definition 1. Let be the subspace ofwhere the elements and are replaced by the single element .
We note that this element is similar to [11], but the number of DOFs is reduced by 2 on each element.
Then, the dimension of is . For , we have , whereThis has 33 unknowns in each element, in which 3 less than the RTN space [5] (see Figure 1).
To define the degrees of freedom, we need an auxiliary space. Let be the subspace consisting of element type :where the elements and are replaced by the single element .
For any , the degrees of freedom are given on face with unit normal and in the interior of as follows:Then, the number of conditions is . We start our analysis of this element by showing that the element is unisolvent.

Figure 1 

DOFs when , 4 for vertical faces and 3 for horizontal faces.

Theorem 1. A vector function is uniquely determined by the degrees of freedom (14)–(17).

Proof. We first note that the dimension of is , and this is also the number of degrees of freedom. Hence, it suffices to prove that if all the conditions are zero, then . Since on the vertical faces, (15) implies for each vertical face. Then, we haveChoosing in (17) shows that . Hence, .
Now, consider . We note thatfor some and . Then, the degrees of freedom (16) impliesFrom (14), we prove that , and we are done.

Definition 2. For the scalar variable, we defineThe dimension of is . Also, we know thatSince , we cannot guarantee that the bilinear forms and make sense for functions of . So, we define extensions to the larger space . LetWe equip with a norm which is an extension of :For , we equip -norm. Now, we have the MFE problem corresponding to (6): find such thatIt is easy to show that the bilinear forms and satisfy the following estimate:for some positive constants and .
Given a typical box element , its boundary consists of six planar pieces which we distinguish into two kinds: the horizontal plane by and the vertical plane . We define an interpolation operator byThen, we have the following lemma.

Lemma 1. If is the interpolation of , then we have

Proof. First, let . Then, we know that for each horizontal face and for each vertical face . Also, we see thatSince and , by the definition of and that and by (29)–(31), we haveLet be an operator from to defined by , for all and . Similarly, we define by for all and .
To confirm the stability of discrete problem (25), we letBy the definitions and the fact , we see that , which leads to the following lemma.

Lemma 2. We havefor some positive constant independent of .
By Lemma 1, the following inf-sup condition can be shown by standard technique [12].

Lemma 3. We have

Theorem 2. Problem (25) has a unique solution.

Proof. The result follows from Lemmas 2 and 3 and Babuka-Brezzi theory.

4. Error Estimates

To prove convergence of our method, we need to estimate the approximation error and the consistency error. Since and have polynomials of degree , we can easily obtain the following approximation error of order :

However, to prove the consistency error is more difficult. We postpone it until later.

To obtain error estimate, we need the following theorem which is essentially given in [10], but we include it for the completeness.

Theorem 3. Let be the solution of (6) and be the discrete solution of (25). Then, we obtain the following error estimates:

Proof. For an arbitrary , we let so thatThen, satisfiesHence, , and we have the following inequality from the coercivity (Lemma 2) of on and (25):for any . We know that both bilinear forms are bounded. From this, we getTherefore,From (35) and (38), we also haveBy (42) and (43), we have the following estimate:Using the first equation of (25), we know thatThen,From (35), we obtainJoining estimates (44) and (47), we obtainwhere is a constant depending only on , and .
Now, we state an assumption regarding the consistency errors:We prove the error estimates.