A Framework for Nonconforming Mixed Finite Element Method for Elliptic Problems in R3

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 nonconformingMFE space satisfying the new patch condition. ,e numerical experiments show that the new MFE shows optimal order convergence in H(div) and L2-norm for various problems with discontinuous coefficient case.


Introduction
e finite element method has achieved great success in many fields, and it has become a powerful tool for solving partial differential equations [1][2][3].
e 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.
ere 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][5][6]. e 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]. e nonconforming approaches have been widely studied for Lagrangian finite elements. So far, all the wellknown mixed finite element (MFE) space are conforming in the sense that the space is contained in H(div, Ω), defined as the space of all vector functions whose divergence belongs to L 2 (Ω). 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. e 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 R 3 together with a new patch condition. A framework for the convergence is given in Section 4. We give numerical experiments in Section 5.

Model Problem
Given Ω ⊂ R 3 , a simply connected bounded Lipschitz polyhedral domain with connected boundary zΩ, we consider the following second-order elliptic problem: where κ ∈ L ∞ (Ω) is assumed to be uniformly positive definite and bounded. And f is a given function in L 2 (Ω). To write the given equation into a mixed system, we use Darcy's law, u � − κ∇p. en, we can rewrite problem (1) in the following mixed form: (Ω) and W � L 2 (Ω) the usual Sobolev spaces with obvious norms. en, we have the following variational form for (2): For the convenience of the presentation, we let en, saddle point problems (3) and (4) can be expressed simply as follows: where (·, ·) indicates the inner product in L 2 (Ω). If familiar inf-sup condition holds, then problem (6) has a unique solution [12].

Nonconforming Mixed Finite Element Spaces
e main idea of MFEM is solving problem (6) over suitable locally defined finite dimensional spaces. eir construction depends on triangulations T h of Ω. Let Ω � [0, 1] 3 and T h : h > 0 be a family of partitions of Ω into parallelepiped K obtained by uniform division having each side length h.
For the construction of the nonconforming MFE space V h , we require that at is, a function in the space V h is locally in H(div, K) but not in H(div, Ω) over the triangulation T h . 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 D in R 2 or R 3 , let P ℓ (D) be the space of polynomials of total degree ℓ and Q ℓ,m (D) or Q ℓ,m,n (D) be the space of polynomials of degree less or equal to ℓ, m, n, respectively, in each variable.

Patch Conditions.
First, we recall a well-known type of "patch condition": let f be the common face of two adjacent elements K i and K j .
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. With hypothesis (H1) holding for q ∈ P ℓ (f) is not enough to determine a continuous interpolation over the whole domain Ω. In fact, if such an interpolation exists, then (H1) holds for q ∈ Q ℓ,ℓ (f), which would imply the conformity of the space V h . us, to study a nonconforming MFE, we need a stronger patch condition but not strong enough to make the V h space fully conforming.
So, we suggest new patch conditions (H2): for all vertical For horizontal faces f H � zK i ∩ zK j and all v h ∈ V h , we have is means that the moments up to order ℓ of the discrete velocity are continuous across horizontal interelement boundaries with respect to Q ℓ,ℓ (f V ), and the moments across vertical interelement boundaries are continuous up to Q ℓ,ℓ (f H )\ x k y k only. Now, we introduce a new nonconforming MFE. We denote by Q * ℓ,m,n the set of all polynomials of Q ℓ,m,n except those having the form x ℓ y m z p for p � 1, . . . , n.
where the elements (x k+1 y k , 0, 0) and (0, x k y k+1 , 0) are replaced by the single element (x k+1 y k , − x k y k+1 , 0). We note that this element is similar to [11], but the number of DOFs is reduced by 2 on each element.
is 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 Ψ h (K) be the subspace consisting of element type ϕ � (ϕ 1 , ϕ 2 , 0): where the elements (x k− 1 y k , 0, 0) and (0, x k y k− 1 , 0) are replaced by the single element (x k− 1 y k , − x k y k− 1 , 0). For any u h � (u 1 , u 2 , u 3 ) ∈ V h (K), the degrees of freedom are given on face f with unit normal n and in the interior of K as follows: en, the number of conditions is 2 (k + 1) 2 − 1} + 4(k + 1) 2 + k(k + 1) 2 − (k − 1) + 2k + 2 k(k + 1) 2 − k} − 1. We start our analysis of this element by showing that the element is unisolvent.
Proof. We first note that the dimension of V h (K) is 3k 3 + 12k 2 + 14k + 4, and this is also the number of degrees of freedom. Hence, it suffices to prove that if all the conditions are zero, then u h � 0. Since u h · n ∈ Q k,k (f V ) on the vertical faces, (15) implies u h · n � 0 for each vertical face. en, we have Choosing ϕ � v in (17) shows that v � 0. Hence, for some s ∈ Q * k,k,k− 1 (K) ⊕ span x k+1 y i , x i y k+1 , i � 0,.. ., k − 1} and r 1 , r 2 ∈ Q k,k (f H )\ x k y k . en, the degrees of freedom (16) implies From (14), we prove that u 3 � 0, and we are done. □ Definition 2. For the scalar variable, we define Since V h ⊈ V, we cannot guarantee that the bilinear forms a(·, ·) and b(·, ·) make sense for functions of V h . So, we define extensions to the larger space We equip X h with a norm which is an extension of ‖ · ‖ H(div,Ω) : For W h , we equip L 2 -norm. Now, we have the MFE problem corresponding to (6) It is easy to show that the bilinear forms a h (·, ·) and b h (·, ·) satisfy the following estimate: for some positive constants c 1 and c 2 . Given a typical box element K, its boundary zK consists of six planar pieces which we distinguish into two kinds: the horizontal plane by f H and the vertical plane f V . We define an interpolation operator Π h : (27) en, we have the following lemma.

Lemma 1. If Π h u is the interpolation of u, then we have
Proof. First, let q ∈ W h (K). en, we know that Since (z x q, z y q, 0) ∈ Ψ k (K) and z z q ∈ Q * k,k,k− 1 ⊕ x k+1 y i , x i y k+1 , i � 0, . . . , k − 1 , by the definition of Π h and that q ∈ W h (K) and by (29)-(31), we have

(32)
Let B be an operator from X h to W ′ defined by  B(v, q) � b(v, q), for all v ∈ X h and q ∈ W. Similarly, we define B h : To confirm the stability of discrete problem (25), we let 4 Journal of Applied Mathematics By the definitions and the fact div V h � W h , we see that N(B h ) ⊂ N(B), which leads to the following lemma.
for some positive constant α independent of h. By Lemma 1, the following inf-sup condition can be shown by standard technique [12].

Theorem 2. Problem (25) has a unique solution.
Proof. e result follows from Lemmas 2 and 3 and Babus

Error Estimates
To prove convergence of our method, we need to estimate the approximation error and the consistency error. Since V h and W h have polynomials of degree k, we can easily obtain the following approximation error of order k: 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 (u, p) ∈ V × W be the solution of (6) and (u h , p h ) ∈ V h × W h be the discrete solution of (25). en, we obtain the following error estimates: Proof. For an arbitrary en, Hence, u h − x h ∈ N(B h ), and we have the following inequality from the coercivity (Lemma 2) of a h (·, ·) on N(B h ) and (25): We bound I by the Cauchy-Schwarz inequality and interpolation property: Similarly, we can bound II as Combining inequalities (54) and (55), we have By dividing ‖v h ‖ H(div,Ω) and taking supremum over v h , we obtain (A.1).

Numerical Results
In this section, we report some results of the numerical experiment using our new element for k � 1. We remark that the optimal convergence rate expected with respective eorem 4 is when p ∈ H 3 (Ω) and u ∈ H 2 (Ω). We solve problem (2) Tables 1 and 2. We see that we obtain optimal convergence for both examples.
Example 1. We choose κ � 1 for 0 < z < (1/2) and κ � 100 is chosen as the exact solution. We report L 2 -errors for the pressure variable and H(div)-error for the velocity variable in Table 1. We observe optimal convergence in both variables.

Conclusion
MFEMs to solve for elliptic equations have been actively studied because approximating the Darcy velocity is as significant as approximating the displacement variable. Most of the MFEMs proposed so far were based on the conforming method in the sense that the space of all vector functions is contained in H(div). On the contrary, the nonconforming approach-based MFEMs were barely studied. In this paper, we suggest a new patch condition under which the nonconforming MFEM has optimal convergence and provides a framework for the convergence. Also, we introduce a new family of nonconforming MFE space which satisfies this condition and provide numerical experiments supporting our analysis.

Data Availability
e datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.