# A Framework for Nonconforming Mixed Finite Element Method for Elliptic Problems inℝ3

Authors: Gwanghyun Jo; J. H. Kim
Journal: Journal of Applied Mathematics
 (2020)

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## 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 inHdiv and L2-norm for various problems with discontinuous coefficient case.


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## Body

## 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 inHdiv,Ω, defined as the space of all vector functions whose divergence belongs to L2Ω. 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 Section3, we introduce nonconforming mixed finite element spaces on parallelepiped in ℝ3 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

GivenΩ⊂ℝ3, a simply connected bounded Lipschitz polyhedral domain with connected boundary ∂Ω, we consider the following second-order elliptic problem:(1)−divκ∇p=f,inΩ,p=0,on∂Ω,where κ∈L∞Ω is assumed to be uniformly positive definite and bounded. And f is a given function in L2Ω. To write the given equation into a mixed system, we use Darcy’s law, u=−κ∇p. Then, we can rewrite problem (1) in the following mixed form:(2)u+κ∇p=0,inΩ,divu=f,inΩ,p=0,on∂Ω.Denote byV=Hdiv,Ω=v∈L2Ω3:divv∈L2Ω and W=L2Ω the usual Sobolev spaces with obvious norms. Then, we have the following variational form for (2): find u,p∈V×W such that(3)∫Ωκ−1uvdx−∫Ωpdivvdx=0,∀v∈V,(4)∫Ωdivu⋅qdx=∫Ωfqdx,∀q∈W.For the convenience of the presentation, we let(5)au,v=∫Ωκ−1uvdx,bv,p=−∫Ωpdivvdx.Then, saddle point problems (3) and (4) can be expressed simply as follows: find u,p∈V×W such that(6)au,v+bv,p=0,∀v∈V,bu,q=−f,q,∀q∈W,where ⋅,⋅ indicates the inner product in L2Ω. 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 Th of Ω. Let Ω=0,13 and Th: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 spaceVh, we require that(7)1Vh=v∈L2Ω3vK∈Hdiv,K,∀K∈Th,2Vh⊈V=Hdiv,Ω.That is, a function in the spaceVh is locally in Hdiv,K but not in Hdiv,Ω over the triangulation Th. 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 domainD in ℝ2 or ℝ3, let PℓD be the space of polynomials of total degree ℓ and Qℓ,mD or Qℓ,m,nD be the space of polynomials of degree less or equal to ℓ,m,n, respectively, in each variable.


### 3.1. Patch Conditions

First, we recall a well-known type of “patch condition”: letf be the common face of two adjacent elements Ki and Kj.(8)H1∫fqvhKi⋅n−vhKj⋅ndA=0,∀q∈Pℓffor someℓ=1,2,…,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∈Pℓf 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∈Qℓ,ℓf, which would imply the conformity of the space Vh. Thus, to study a nonconforming MFE, we need a stronger patch condition but not strong enough to make the Vh space fully conforming.So, we suggest new patch conditions (H2): for all vertical faces fV:=∂Ki∩∂Kji≠j and all vh∈Vh, we have(9)H2a∫fVqvhKi⋅n−vhKj⋅ndA=0,∀q∈Qℓ,ℓfV.For horizontal facesfH=∂Ki∩∂Kj and all vh∈Vh, we have(10)H2b∫fHqvhKi⋅n−vhKj⋅ndA=0,∀q∈Qℓ,ℓfH\xkyk.This means that the moments up to orderℓ of the discrete velocity are continuous across horizontal interelement boundaries with respect to Qℓ,ℓfV, and the moments across vertical interelement boundaries are continuous up to Qℓ,ℓfH\xkyk only.Now, we introduce a new nonconforming MFE. We denote byQℓ,m,n∗ the set of all polynomials of Qℓ,m,n except those having the form xℓymzp for p=1,…,n.Definition 1.
LetVhK be the subspace of(11)Qk+1,k,k∗K×Qk,k+1,k∗K×Qk,k,k+1∗K⊕xk+1yi,xiyk+1,i=0,…,k−1,where the elements xk+1yk,0,0 and 0,xkyk+1,0 are replaced by the single element xk+1yk,−xkyk+1,0.
We note that this element is similar to [11], but the number of DOFs is reduced by 2 on each element.
Then, the dimension ofVhK is 2k+2k+12−k+k+12k+2−k+1+2k−1=3k3+12k2+14k+4. For k=1, we have u=u1,u2,u3, where(12)u1=P1x,y,z+a1xy+a2yz+a3zx+a4xyz+a5x2+a6x2z+dx2y,u2=P1x,y,z+b1xy+b2yz+b3zx+b4xyz+b5y2+b6y2z−dxy2,u3=P1x,y,z+c1xy+c2yz+c3zx+c4z2+c5xz2+c6yz2+c7x2+c8y2.
This 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. LetΨhK be the subspace consisting of element type ϕ=ϕ1,ϕ2,0:(13)Qk−1,k,k∗K×Qk,k−1,k∗K×0,where the elements xk−1yk,0,0 and 0,xkyk−1,0 are replaced by the single element xk−1yk,−xkyk−1,0.
For anyuh=u1,u2,u3∈VhK, the degrees of freedom are given on face f with unit normal n and in the interior of K as follows:(14)∫fHuh⋅nqdA,∀q∈Qk,kfH\xkyk,for each horizontal facefH,(15)∫fVuh⋅nqdA,∀q∈Qk,kfV,for each vertical facefV,(16)∫Ku3ϕ3dx,∀ϕ3∈Qk,k,k−1∗⊕spanxk+1yi,xiyk+1,i=0,…,k−1,(17)∫Ku1ϕ1+u2ϕ2dx,∀ϕ1,ϕ2,0∈ΨhK.
Then, the number of conditions is2k+12−1+4k+12+kk+12−k−1+2k+2kk+12−k−1. We start our analysis of this element by showing that the element is unisolvent.Figure 1
DOFs whenk=1, 4 for vertical faces and 3 for horizontal faces.Theorem 1.
A vector functionuh=u1,u2,u3∈VhK is uniquely determined by the degrees of freedom (14)–(17).Proof.
We first note that the dimension ofVhK is 3k3+12k2+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 uh=0. Since uh⋅n∈Qk,kfV on the vertical faces, (15) implies uh⋅n=0 for each vertical face. Then, we have(18)u1=x1−xv1,u2=y1−yv2,v=v1,v2,0∈ΨhK.
Choosingϕ=v in (17) shows that v=0. Hence, u1=u2=0.
Now, consideru3∈Qk,k,k+1∗K⊕spanxk+1yi,xiyk+1,i=0,…,k−1. We note that(19)u3=s+r1zk+r2zk+1,for some s∈Qk,k,k−1∗K⊕spanxk+1yi,xiyk+1,i=0,…,k−1 and r1,r2∈Qk,kfH\xkyk. Then, the degrees of freedom (16) implies(20)u3=r1′zk+r2′zk+1,∀r1′,r2′∈Qk,kfH\xkyk.
From (14), we prove that u3=0, and we are done.Definition 2.
For the scalar variable, we define(21)WhK=Qk,k,k∗\xkyk.
The dimension ofWhK is k+13−k−1=k3+3k2+2k. Also, we know that(22)1dimVhK=WhK,2Wh⊂W.
SinceVh⊈V, we cannot guarantee that the bilinear forms a⋅,⋅ and b⋅,⋅ make sense for functions of Vh. So, we define extensions to the larger space Xh=Vh+Hdiv,Ω. Let(23)ahu,v=∑i∫Kiκ−1uvdx,∀u,v∈Xh,bhv,p=−∑i∫Kipdivvdx,∀p∈Wh,∀v∈Xh.
We equipXh with a norm which is an extension of ⋅Hdiv,Ω:(24)uXh2=∑iuHdiv,Ki2,∀u∈Xh.
ForWh, we equip L2-norm. Now, we have the MFE problem corresponding to (6): find uh,ph∈Xh×Wh such that(25)ahuh,vh+bhvh,ph=0,∀vh∈Vh,bhuh,qh=−f,qh,∀qh∈Wh.
It is easy to show that the bilinear formsah⋅,⋅ and bh⋅,⋅ satisfy the following estimate:(26)ahuh,vh≤c1uhXhvhXh,bhvh,ph≤c2phWhvhXh,for some positive constants c1 and c2.
Given a typical box elementK, its boundary ∂K consists of six planar pieces which we distinguish into two kinds: the horizontal plane by fH and the vertical plane fV. We define an interpolation operator Πh:Hk+1K⟶VhK by(27)∫fHu−Πhu⋅nqdA=0,∀q∈Qk,kfH\xkyk,∫fVu−Πhu⋅nqdA=0,∀q∈Qk,kfV,∫Ku3−Πhu3ϕ3dx=0,∀ϕ3∈Qk,k,k−1∗⊕spanxk+1yi,xiyk+1,i=0,…,k−1,∫Ku1−Πhu1ϕ1+u2−Πhu2ϕ2dx=0,∀ϕ1,ϕ2,0∈ΨhK.
Then, we have the following lemma.Lemma 1.
IfΠhu is the interpolation of u, then we have(28)∫Kdivu−Πhuqdx=0,∀u∈VhK,q∈WhK.Proof.
First, letq∈WhK. Then, we know that qfH∈Qk,kfH\xkyk for each horizontal face fH and qfV∈Qk,kfV for each vertical face fV. Also, we see that(29)∂xq∈Qk−1,k,k∗\xk−1yk,(30)∂yq∈Qk,k−1,k∗\xkyk−1,(31)∂zq∈Qk,k,k−1∗\xkyk.
Since∂xq,∂yq,0∈ΨkK and ∂zq∈Qk,k,k−1∗⊕xk+1yi,xiyk+1,i=0,…,k−1, by the definition of Πh and that q∈WhK and by (29)–(31), we have(32)∫KdivΠhuqdx=∫∂KΠhu⋅nqdA−∫KΠhu⋅∇qdx=∫∂fHΠhu⋅nqdA+∫∂fVΠhu⋅nqdA−∫KΠhu3∂zqdx+∫KΠhu1∂xq+Πhu2∂yqdx=∫∂fHu⋅nqdA+∫∂fVu⋅nqdA−∫Ku3∂zqdx+∫Ku1∂xq+u2∂yqdx=∫∂Ku⋅nqdA−∫Ku⋅∇qdx=∫Kdivuqdx.
LetB be an operator from Xh to W′ defined by Bv,q=bv,q, for all v∈Xh and q∈W. Similarly, we define Bh:Vh⟶Wh′ by Bhvh,qh=bhv,qh for all vh∈Vh and qh∈Wh.
To confirm the stability of discrete problem (25), we let(33)NBh=vh∈Vhbhvh,qh=0,∀qh∈Wh,NB=v∈Xhdivv|K=0,∀K∈Th,NBh∗=ph∈Whbhvh,ph=0,∀vh∈Vh.
By the definitions and the factdivVh=Wh, we see that NBh⊂NB, which leads to the following lemma.Lemma 2.
We have(34)supvh∈NBhahvh,whvhXh≥αwhXh,∀wh∈NBh,for some positive constant α independent of h.
By Lemma1, the following inf-sup condition can be shown by standard technique [12].Lemma 3.
We have(35)supvh∈Xhbhvh,qhvhXh≥βqhWh/NBh∗,∀qh∈Wh.Theorem 2.
Problem (25) has a unique solution.Proof.
The result follows from Lemmas2 and 3 and Babus⌣ka-Brezzi theory.


## 3.1. Patch Conditions

First, we recall a well-known type of “patch condition”: letf be the common face of two adjacent elements Ki and Kj.(8)H1∫fqvhKi⋅n−vhKj⋅ndA=0,∀q∈Pℓffor someℓ=1,2,…,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∈Pℓf 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∈Qℓ,ℓf, which would imply the conformity of the space Vh. Thus, to study a nonconforming MFE, we need a stronger patch condition but not strong enough to make the Vh space fully conforming.So, we suggest new patch conditions (H2): for all vertical faces fV:=∂Ki∩∂Kji≠j and all vh∈Vh, we have(9)H2a∫fVqvhKi⋅n−vhKj⋅ndA=0,∀q∈Qℓ,ℓfV.For horizontal facesfH=∂Ki∩∂Kj and all vh∈Vh, we have(10)H2b∫fHqvhKi⋅n−vhKj⋅ndA=0,∀q∈Qℓ,ℓfH\xkyk.This means that the moments up to orderℓ of the discrete velocity are continuous across horizontal interelement boundaries with respect to Qℓ,ℓfV, and the moments across vertical interelement boundaries are continuous up to Qℓ,ℓfH\xkyk only.Now, we introduce a new nonconforming MFE. We denote byQℓ,m,n∗ the set of all polynomials of Qℓ,m,n except those having the form xℓymzp for p=1,…,n.Definition 1.
LetVhK be the subspace of(11)Qk+1,k,k∗K×Qk,k+1,k∗K×Qk,k,k+1∗K⊕xk+1yi,xiyk+1,i=0,…,k−1,where the elements xk+1yk,0,0 and 0,xkyk+1,0 are replaced by the single element xk+1yk,−xkyk+1,0.
We note that this element is similar to [11], but the number of DOFs is reduced by 2 on each element.
Then, the dimension ofVhK is 2k+2k+12−k+k+12k+2−k+1+2k−1=3k3+12k2+14k+4. For k=1, we have u=u1,u2,u3, where(12)u1=P1x,y,z+a1xy+a2yz+a3zx+a4xyz+a5x2+a6x2z+dx2y,u2=P1x,y,z+b1xy+b2yz+b3zx+b4xyz+b5y2+b6y2z−dxy2,u3=P1x,y,z+c1xy+c2yz+c3zx+c4z2+c5xz2+c6yz2+c7x2+c8y2.
This 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. LetΨhK be the subspace consisting of element type ϕ=ϕ1,ϕ2,0:(13)Qk−1,k,k∗K×Qk,k−1,k∗K×0,where the elements xk−1yk,0,0 and 0,xkyk−1,0 are replaced by the single element xk−1yk,−xkyk−1,0.
For anyuh=u1,u2,u3∈VhK, the degrees of freedom are given on face f with unit normal n and in the interior of K as follows:(14)∫fHuh⋅nqdA,∀q∈Qk,kfH\xkyk,for each horizontal facefH,(15)∫fVuh⋅nqdA,∀q∈Qk,kfV,for each vertical facefV,(16)∫Ku3ϕ3dx,∀ϕ3∈Qk,k,k−1∗⊕spanxk+1yi,xiyk+1,i=0,…,k−1,(17)∫Ku1ϕ1+u2ϕ2dx,∀ϕ1,ϕ2,0∈ΨhK.
Then, the number of conditions is2k+12−1+4k+12+kk+12−k−1+2k+2kk+12−k−1. We start our analysis of this element by showing that the element is unisolvent.Figure 1
DOFs whenk=1, 4 for vertical faces and 3 for horizontal faces.Theorem 1.
A vector functionuh=u1,u2,u3∈VhK is uniquely determined by the degrees of freedom (14)–(17).Proof.
We first note that the dimension ofVhK is 3k3+12k2+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 uh=0. Since uh⋅n∈Qk,kfV on the vertical faces, (15) implies uh⋅n=0 for each vertical face. Then, we have(18)u1=x1−xv1,u2=y1−yv2,v=v1,v2,0∈ΨhK.
Choosingϕ=v in (17) shows that v=0. Hence, u1=u2=0.
Now, consideru3∈Qk,k,k+1∗K⊕spanxk+1yi,xiyk+1,i=0,…,k−1. We note that(19)u3=s+r1zk+r2zk+1,for some s∈Qk,k,k−1∗K⊕spanxk+1yi,xiyk+1,i=0,…,k−1 and r1,r2∈Qk,kfH\xkyk. Then, the degrees of freedom (16) implies(20)u3=r1′zk+r2′zk+1,∀r1′,r2′∈Qk,kfH\xkyk.
From (14), we prove that u3=0, and we are done.Definition 2.
For the scalar variable, we define(21)WhK=Qk,k,k∗\xkyk.
The dimension ofWhK is k+13−k−1=k3+3k2+2k. Also, we know that(22)1dimVhK=WhK,2Wh⊂W.
SinceVh⊈V, we cannot guarantee that the bilinear forms a⋅,⋅ and b⋅,⋅ make sense for functions of Vh. So, we define extensions to the larger space Xh=Vh+Hdiv,Ω. Let(23)ahu,v=∑i∫Kiκ−1uvdx,∀u,v∈Xh,bhv,p=−∑i∫Kipdivvdx,∀p∈Wh,∀v∈Xh.
We equipXh with a norm which is an extension of ⋅Hdiv,Ω:(24)uXh2=∑iuHdiv,Ki2,∀u∈Xh.
ForWh, we equip L2-norm. Now, we have the MFE problem corresponding to (6): find uh,ph∈Xh×Wh such that(25)ahuh,vh+bhvh,ph=0,∀vh∈Vh,bhuh,qh=−f,qh,∀qh∈Wh.
It is easy to show that the bilinear formsah⋅,⋅ and bh⋅,⋅ satisfy the following estimate:(26)ahuh,vh≤c1uhXhvhXh,bhvh,ph≤c2phWhvhXh,for some positive constants c1 and c2.
Given a typical box elementK, its boundary ∂K consists of six planar pieces which we distinguish into two kinds: the horizontal plane by fH and the vertical plane fV. We define an interpolation operator Πh:Hk+1K⟶VhK by(27)∫fHu−Πhu⋅nqdA=0,∀q∈Qk,kfH\xkyk,∫fVu−Πhu⋅nqdA=0,∀q∈Qk,kfV,∫Ku3−Πhu3ϕ3dx=0,∀ϕ3∈Qk,k,k−1∗⊕spanxk+1yi,xiyk+1,i=0,…,k−1,∫Ku1−Πhu1ϕ1+u2−Πhu2ϕ2dx=0,∀ϕ1,ϕ2,0∈ΨhK.
Then, we have the following lemma.Lemma 1.
IfΠhu is the interpolation of u, then we have(28)∫Kdivu−Πhuqdx=0,∀u∈VhK,q∈WhK.Proof.
First, letq∈WhK. Then, we know that qfH∈Qk,kfH\xkyk for each horizontal face fH and qfV∈Qk,kfV for each vertical face fV. Also, we see that(29)∂xq∈Qk−1,k,k∗\xk−1yk,(30)∂yq∈Qk,k−1,k∗\xkyk−1,(31)∂zq∈Qk,k,k−1∗\xkyk.
Since∂xq,∂yq,0∈ΨkK and ∂zq∈Qk,k,k−1∗⊕xk+1yi,xiyk+1,i=0,…,k−1, by the definition of Πh and that q∈WhK and by (29)–(31), we have(32)∫KdivΠhuqdx=∫∂KΠhu⋅nqdA−∫KΠhu⋅∇qdx=∫∂fHΠhu⋅nqdA+∫∂fVΠhu⋅nqdA−∫KΠhu3∂zqdx+∫KΠhu1∂xq+Πhu2∂yqdx=∫∂fHu⋅nqdA+∫∂fVu⋅nqdA−∫Ku3∂zqdx+∫Ku1∂xq+u2∂yqdx=∫∂Ku⋅nqdA−∫Ku⋅∇qdx=∫Kdivuqdx.
LetB be an operator from Xh to W′ defined by Bv,q=bv,q, for all v∈Xh and q∈W. Similarly, we define Bh:Vh⟶Wh′ by Bhvh,qh=bhv,qh for all vh∈Vh and qh∈Wh.
To confirm the stability of discrete problem (25), we let(33)NBh=vh∈Vhbhvh,qh=0,∀qh∈Wh,NB=v∈Xhdivv|K=0,∀K∈Th,NBh∗=ph∈Whbhvh,ph=0,∀vh∈Vh.
By the definitions and the factdivVh=Wh, we see that NBh⊂NB, which leads to the following lemma.Lemma 2.
We have(34)supvh∈NBhahvh,whvhXh≥αwhXh,∀wh∈NBh,for some positive constant α independent of h.
By Lemma1, the following inf-sup condition can be shown by standard technique [12].Lemma 3.
We have(35)supvh∈Xhbhvh,qhvhXh≥βqhWh/NBh∗,∀qh∈Wh.Theorem 2.
Problem (25) has a unique solution.Proof.
The result follows from Lemmas2 and 3 and Babus⌣ka-Brezzi theory.


## 4. Error Estimates

To prove convergence of our method, we need to estimate the approximation error and the consistency error. SinceVh and Wh have polynomials of degree k, we can easily obtain the following approximation error of order k:(36)infvh∈Vhu−vhVh+infqh∈Whp−qhWh≤Chk+1pHk+2Ω.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.
Letu,p∈V×W be the solution of (6) and uh,ph∈Vh×Wh be the discrete solution of (25). Then, we obtain the following error estimates:(37)u−uhXh+p−phθWh≤Cinfvh∈Vhu−vhVh+infqh∈Whp−qhWh+supvh∈Vhahu,vh+bhvh,pvhXh.Proof.
For an arbitraryvh∈Vh, we let wh=Πhu−vh so that(38)bhwh,qh=bhu−vh,qh,∀qh∈Wh.
Then,xh=vh+wh satisfies(39)bhxh,qh=bhu,qh,∀qh∈Wh.
Hence,uh−xh∈NBh, and we have the following inequality from the coercivity (Lemma 2) of ah⋅,⋅ on NBh and (25):(40)αuh−xhXh≤supvh∈NBh1vhXhahvh,uh−xh≤supvh∈NBh1vhXhahvh,uh−u+ahvh,u−xh≤supvh∈NBh1vhXhahvh,u−xh−bhvh,qh−p−ahu,vh+bhvh,p,for any qh∈Wh. We know that both bilinear forms are bounded. From this, we get(41)αuh−xhX≤c1u−xhX+c2p−qhW+supvh∈NBhahu,vh+bhvh,pvhX,αu−uhXh≤αu−xhXh+uh−xhXh=α+c1u−xhX+c2p−qhW+supvh∈NBhahu,vh+bhvh,pvhX.
Therefore,(42)u−uhXh≤1+c1αu−xhXh+c2αp−qhXh+1αsupvh∈NBhahu,vh+bhvh,pvhXh.
From (35) and (38), we also have(43)u−xhXh≤u−vhXh+whXh≤1+c2βu−vhXh.
By (42) and (43), we have the following estimate:(44)u−uhXh≤1+c1α1+c2βu−vhXh+c2αp−qhXh+1αsupvh∈NBhahu,vh+bhvh,pvhXh.
Using the first equation of (25), we know that(45)ahu−uh,vh+bhvh,p−ph=ahu,vh+bhvh,p,∀vh∈Vh.
Then,(46)bhvh,p−ph=−au−uh,vh+ahu,vh+bhvh,p,∀vh∈Vh.
From (35), we obtain(47)p−phWh/NBh∗≤1βsupvh∈Vhbhvh,p−phvhXh≤1βc3u−uhXh+supvh∈Vhahu,vh+bhvh,pvhXh.
Joining estimates (44) and (47), we obtain(48)u−uhXh+p−phWh≤Cinfvh∈Vhu−vhVh+infqh∈Whp−qhWh+supvh∈Vhahu,vh+bhp,vhvhXh,where C is a constant depending only on c1,c2,c3,α, and β.
Now, we state an assumption regarding the consistency errors:(49)A.1supvh∈Vhahu,vh+bhvh,pvhXh≤ChℓpHk+2Ω,ℓ=korℓ=k+1.
We prove the error estimates.Theorem 4.
Suppose the following regularity holds: the solutionu,p of (6) satisfies u∈Hk+1Ω, p∈Hk+2Ω, and there exists some constant C>0 such that(50)uk+1+pk+2≤CfHkΩ.
If (A.1) is satisfied, then the following estimates hold:(51)u−uhXh+p−phWh≤Chmink+1,ℓpHk+2Ω≤Chmink+1,ℓfHkΩ.Proof.
This follows directly from Theorem3, Assumption (A.1), and (36).
Now, we prove (A.1) for the caseℓ=k. The case ℓ=k+1 is left open.Proof of (A.1).
By the integration by parts, we see that(52)ahu,vh+bhp,vh=∑K∈Th∫Kκ−1uvhdx−∑K∈Th∫Kpdivvhdx=∑K∈Th∫Kκ−1uvhdx−∑K∈Th∫∂Kpvh⋅ndσ+∑K∈Th∫K∇p⋅vhdx=−∑K∈Th∫∂Kpvh⋅ndσ=−∑K∈Th∫Kdivpvhdx.
Letlhp be a continuous interpolation of p such that lhpfV∈Qk,k and K^hpfH∈Qk,k\xkyk. Then, by the patch condition (H2(a), H2(b)), we see(53)∑K∈Th∫Kdivpvhdx=∑K∈Th∫∂Kpvh⋅ndσ=∑K∈Th∫∂Kp−lhpvh⋅ndσ=∑K∈Th∫Kdivp−lhpvhdx=∑K∈Th∫K∇p−lhp⋅vhdx+∑K∈Th∫Kp−lhpdivvhdx=I+II.
We bound I by the Cauchy–Schwarz inequality and interpolation property:(54)I≤p−lhpL2ΩvhHdiv,Ω≤Chk+1pHk+2ΩvhHdiv,Ω.
Similarly, we can boundII as(55)II≤∇p−lhpL2ΩvhL2Ω≤ChkpHk+2ΩvhL2Ω.
Combining inequalities (54) and (55), we have(56)ahu,vh+bhp,vh≤ChkpHk+2ΩvhHdiv,Ω.
By dividingvhHdiv,Ω and taking supremum over vh, we obtain (A.1).


## 5. Numerical Results

In this section, we report some results of the numerical experiment using our new element fork=1. We remark that the optimal convergence rate expected with respective Theorem 4 is(57)u−uhHdiv,Ω+p−phL2Ω≤Ch2,when p∈H3Ω and u∈H2Ω. We solve problem (2) on the unit cube Ω=0,13. The numerical experiments are carried out on uniform triangulation Th by cubes having size h=2−m, m=1,2,…,5. We consider two examples where the coefficients κ are discontinuous across some interfaces. In each example, the pressure (respectively, velocity) variable belongs to H3 (respectively, H2) on each subregion, and we report L2-errors for the pressure variable and Hdiv-error for the velocity variable in Tables 1 and 2. We see that we obtain optimal convergence for both examples.Table 1
L
2-error for pressure and Hdiv-error of velocity (Example 1).
Level
p
−
p
h
L
2
Order
u
−
u
h
H
div
Order
1
3.652E − 04
1.649E − 02
2
1.043E − 04
1.807
4.626E − 03
1.834
3
2.681E − 05
1.960
1.188E − 03
1.961
4
6.746E − 06
1.991
2.990E − 04
1.990
5
1.689E − 06
1.998
7.488E − 05
1.998Table 2
L
2-error for pressure and Hdiv-error of velocity (Example 2).
Level
p
−
p
h
L
2
Order
u
−
u
h
H
div
Order
1
1.324E − 05
2.608E − 03
2
5.113E − 06
1.373
8.617E − 04
1.597
3
1.469E − 06
1.799
2.327E − 04
1.889
4
3.802E − 07
1.950
5.932E − 05
1.972
5
9.587E − 08
1.988
1.490E − 05
1.993Example 1.
We chooseκ=1 for 0<z<1/2 and κ=100 for 1/2≤z<1. The function(58)px,y,z=1κx1−xy1−yzz−0.51−zis chosen as the exact solution. We report L2-errors for the pressure variable and H(div)-error for the velocity variable in Table 1. We observe optimal convergence in both variables.Example 2.
We divideΩ=0,13 by eight octant Ωi,s, i=1,2,3,4 and s=t,b (see Figure 2). Here, Ωi,b are the four bottom regions z≤0.5 and Ωi,t are four top regions z>0.5.
The coefficientκ is given by(59)κ=1,forx∈Ω1,b,10,forx∈Ω2,b,100,forx∈Ω3,b,1000,forx∈Ω4,b,κ=10,forx∈Ω1,t,100,forx∈Ω2,t,1000,forx∈Ω3,t,1,forx∈Ω4,t,and p=1/κLv, where L=x−0.5y−0.5z−0.5 and v=x1−xy1−yz1−z. We report the errors in Table 2. Again, we observe optimal convergence in both variables.Figure 2
SubregionsΩi,b and Ωi,t, i=1,2,3,4.


## 6. 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 inHdiv. 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.



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