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This paper seeks to extend these ideas to the mixed formulation of second order boundary value problems. A "patch--test" is designed as a powerful tool for probing the viability of nonconforming approximations. In particular the criteria are utterly problem--independent. Consequently they can dispense with any regularity requirements. Key words. mi...
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... 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. ...
... where n 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. ...
... To obtain error estimate, we need the following theorem which is essentially given in [10], but we include it for the completeness. ...
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.
In a recent work, Hiptmair [Mathematisches Institut, M9404, 1994] has constructed and analyzed a family of nonconforming mixed finite elements for second-order elliptic problems. However, his analysis does not work on the lowest order elements. In this article, we show that it is possible to construct a nonconforming mixed finite element for the lowest order case. We prove the convergence and give estimates of optimal order for this finite element. Our proof is based on the use of the properties of the so-called nonconforming bubble function to control the consistency terms introduced by the nonconforming approximation. We further establish an equivalence between this mixed finite element and the nonconforming piecewise quadratic finite element of Fortin and Soulie [J. Numer. Methods Eng., 19, 505–520, 1983]. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 445–457, 1997
In this article, we introduce three schemes for the Poisson problem in 2D on triangular meshes, generalizing the FVbox scheme introduced by Courbet and Croisille [1]. In this kind of scheme, the approximation is performed on the mixed form of the problem, but contrary to the standard mixed method, with a pair of trial spaces different from the pair of test spaces. The latter is made of Galerkin-discontinuous spaces on a unique mesh. The first scheme uses as trial spaces the P1 nonconforming space of Crouzeix-Raviart both for u and for the flux p = ∇u. In the two others, the quadratic nonconforming space of Fortin and Soulie is used. An important feature of all these schemes is that they are equivalent to a first scheme in u only and an explicit representation formula for the flux p = ∇u. The numerical analysis of the schemes is performed using this property. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 355–373, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10003
The main aim of this paper is to develop a new low-order non-conforming mixed finite-element method to solve the second-order elliptic problems on rectangular meshes. The convergence analysis is presented and the optimal error estimates are derived with the lowest regularity of the exact solution. Numerical results which coincide with our theoretical analysis show that this element indeed has very good convergence behaviour.
