 # Piecewise Divergence-Free H(div)-Nonconforming Virtual Elements for Stokes Problem in Any Dimensions

Piecewise divergence-free H(div)-nonconforming virtual elements are designed for Stokes problem in any dimensions. After introducing a local energy projector based on the Stokes problem and the stabilization, a divergence-free nonconforming virtual element method is proposed for Stokes problem. A detailed and rigorous error analysis is presented for the discrete method, including the norm equivalence of the stabilization on the kernel of the local energy projector, the interpolation error estimate, the discrete inf-sup condition, and the optimal error estimate of the discrete method. An important property in the analysis is that the local energy projector commutes with the divergence operator. A reduced virtual element method is also discussed. Numerical results are provided to verify the theoretical convergence.

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## 1. Introduction

In this paper, we shall construct piecewise divergence-free -nonconforming virtual elements for Stokes problem in any dimensions. Assume that is a bounded polytope. The Stokes problem is governed by

 (1.1) ⎧⎨⎩−div(νε(u))−∇p=f in Ω,divu=0 in Ω,u=0 on ∂Ω,

where is the velocity field, is the pressure, is the symmetric gradient of , external force field , and constant is the viscosity. The incompressibility constraint in (1.1) describes the conservation of mass for the incompressible fluid.

Since the nonconforming - element is a stable pair for the Stokes problem , as the generalization of the nonconforming element, it is spontaneous that the -nonconforming virtual element in  is adopted to discretize the Stokes problem in [13, 25]. While the incompressibility constraint is not satisfied exactly in general at the discrete level for the discrete methods in [13, 25], which is very important for the Brinkman problem  and the Navier-Stokes problem . To design the discrete method with the exact divergence-free discrete velocity, one idea is to combine the discontinuous Galerkin technique and the -conforming finite elements or virtual elements, such as the discontinuous Galerkin -conforming method  and the divergence-free weak virtual element method . The more compact idea in [7, 6, 2] is to construct divergence-free conforming virtual elements in two and three dimensions by defining the space of shape functions through the local Stokes problem with Dirichlet boundary condition. By enriching an -conforming virtual element with some divergence-free functions, a divergence-free nonconforming virtual element in two dimensions is advanced in , in which each element in the partition is required to be convex. We refer to  for a virtual element method based on the pseudostress-velocity formulation.

Following the ideas in [15, 22], we shall devise piecewise divergence-free -nonconforming virtual elements in any dimensions based on the generalized Green’s identity for Stokes problem, which are also

-nonconforming. The degrees of freedom of the proposed virtual elements for the velocity are same as those in

, i.e. copies of the degrees of freedom of the -nonconforming virtual elements in . And the space of shape functions for the velocity is defined from the local Stokes problem with Neumann boundary condition, which is different from that in  due to the constraint on the boundary. Our virtual elements are locally divergence-free since .

A novelty of this paper is to introduce a local energy projector based on the Stokes problem:

 (ε(ΠKkw),ε(v))K+(divv,PKw)K =(ε(w),ε(v))K∀ v∈Pk(K;Rd), (div(ΠKkw),q)K =(divw,q)K∀ q∈Pk−1(K),

while the local projector is adopted in all the previous papers. The local Stokes-based projector commutes with the divergence operator, i.e.

 (1.2) div(w−ΠKkw)=0∀ w∈Vk(K).

Then we define a stabilization involving all the degrees of freedom of the virtual elements for the velocity except those corresponding to . With the help of the local projector and the stabilization, we propose a piecewise divergence-free nonconforming virtual element method for Stokes problem, where the velocity is discretized by the virtual elements and the pressure is discretized by the piecewise polynomials. Furthermore, applying the technique in [7, 17, 26], we remove the degrees of freedom corresponding to for the velocity, reduce the space of shape functions to , and then derive the reduced virtual element method, in which the pressure is discretized by the piecewise constant. Hence we can first acquire the discrete velocity by solving the reduced discrete method, and then recover the discrete pressure elementwisely.

A detailed and rigorous error analysis is presented for the piecewise divergence-free nonconforming virtual element method. We first prove the inverse inequality

 (1.3) hK∥divε(v)+∇s∥0,K≲∥ε(v)∥0,K∀ v∈Vk(K)

by using the fact that is a norm on the finite-dimensional space , where satisfies . Then we derive the norm equivalence of the stabilization on the kernel of the local projector from (1.2) and (1.3). The interpolation error estimate is acquired after setting up the Galerkin orthogonality of the interpolation operator. With the norm equivalence of the stabilization and the interpolation error estimate, we build up the discrete inf-sup condition, and thus the piecewise divergence-free nonconforming virtual element method is wellposed. Finally the optimal error estimate comes from the discrete inf-sup condition and the interpolation error estimate in a standard way.

The rest of this paper is organized as follows. In Section 2, we present some notation and inequalities. The divergence-free nonconforming virtual elements, local energy projector, stabilization and interpolation operator are constructed in Section 3. We show the divergence-free nonconforming virtual element method for the Stokes problem and its error analysis in Section 4. A reduced virtual element method is given in Section 5. In Section 6, numerical results are provided to verify the theoretical convergence.

## 2. Preliminaries

### 2.1. Notation

Denote by the space of all tensors, the space of all symmetric tensors, and

the space of all skew-symmetric

tensors. Denote the deviatoric part and the trace of the tensor by and accordingly, then we have

 devτ=τ−1d(trτ)I.

Given a bounded domain and a non-negative integer , let be the usual Sobolev space of functions on , and

be the usual Sobolev space of functions taking values in the finite-dimensional vector space

for being , , or . The corresponding norm and semi-norm are denoted respectively by and . Let be the standard inner product on or . If is , we abbreviate , and by , and , respectively. Let be the closure of with respect to the norm . For integer , notation stands for the set of all polynomials over with the total degree no more than . Set . And denote by the vectorial or tensorial version space of . Let () be the -orthogonal projector onto ().

Let be a family of partitions of into nonoverlapping simple polytopal elements with and . Let be the set of all -dimensional faces of the partition for . Moreover, we set for each

 Fr(K):={F∈Frh:F⊂∂K}.

Similarly, for , we define

 F1(F):={e∈F2h:e⊂¯¯¯¯F}.

For any , denote by its diameter and fix a unit normal vector . For any , denote by the unit outward normal to . Without causing any confusion, we will abbreviate as for simplicity.

For non-negative integer , let

 Pk(Th):={v∈L2(Ω):v|K∈Pk(K) for each K∈Th}.

Define

 H1(Th;Rd):={v∈L2(Ω;Rd):v|K∈H1(K;Rd) for each K∈Th},

and the usual broken -type norm and semi-norm

 ∥v∥1,h:=(∑K∈Th∥v∥21,K)1/2,|v|1,h:=(∑K∈Th|v|21,K)1/2.

Let and be the piecewise counterparts of and with respect to .

We introduce jumps on ()-dimensional faces. Consider two adjacent elements and sharing an interior ()-dimensional face . Denote by and the unit outward normals to the common face of the elements and , respectively. For a scalar-valued or tensor-valued function , write and . Then define the jump on as follows:

 ⟦v⟧:=v+nF⋅n++v−nF⋅n−.

On a face lying on the boundary , the above term is defined by

Denote the space of rigid motions by

 RM:={c+Ax:c∈Rd,A∈K},

where . For any , is defined by

 (curlv)ij:=∂vi∂xj−∂vj∂xi for i,j=1,⋯,d.

For positive integer , set . Take being any subspace of such that

 (2.1) Pk−2(K;Rd)=G⊕k−2(K)⊕Gk−2(K),

where is the direct sum. One choice of is given by (3.11) in [3, 4]

 (2.2)

where and is the exterior product. We can also take

 (2.3) G⊕k−2(K)={(x−xK)⊥Pk−3(K), for d=2,(x−xK)∧Pk−3(K;R3), for d=3,

where is the centroid of . Let be the -orthogonal projector onto .

### 2.2. Mesh conditions and some inequalities

We impose the following conditions on the mesh in this paper:

• Each element is star-shaped with respect to a ball with radius , where the chunkiness parameter is uniformly bounded;

• There exists a quasi-uniform simplicial mesh such that each is a union of some simplexes in .

Throughout this paper, we use “” to mean that “”, where is a generic positive constant independent of the mesh size and the viscosity , but may depend on the chunkiness parameter of the polytope, the degree of polynomials , the dimension of space , and the shape regularity and quasi-uniform constants of the virtual triangulation , which may take different values at different appearances. And means and .

Under the mesh condition (A1), we have the trace inequality of [11, (2.18)]

 (2.4) ∥v∥20,∂K≲h−1K∥v∥20,K+hK|v|21,K∀ v∈H1(K),

the Poincaré-Friedrichs inequality [11, (2.15)]

 (2.5) ∥v∥0,K≲hK|v|1,K+h1−d/2K∣∣∣∫∂Kvds∣∣∣∀ v∈H1(K),

and the Korn’s second inequality [23, 1]

 (2.6) |v|1,K≲∥ε(v)∥0,K∀ v∈H1(K;Rd)% satisfying QK0(curlv)=0.

Recall the Babuška-Aziz inequality [19, 8, 20]: for any , there exists such that

 (2.7) divv=q,h−1K∥v∥0,K+|v|1,K≲∥q∥0,K.

When , we can choose . Combined with the proof of Proposition 9.1.1 in , it holds for any satisfying that [14, Lemma 3.4]

 (2.8) ∥τ∥0,K≲∥devτ∥0,K+∥divτ∥−1,K.

Let be the regular inscribed simplex of , where all the edges of share the common length. It holds for any nonnegative integers and that [22, Lemma 4.3 and Lemma 4.4]

 (2.9) ∥q∥0,K≂∥q∥0,Ks∀ q∈Pℓ(K),
 (2.10) ∥q∥0,K≲h−iK∥q∥−i,K∀ q∈Pℓ(K).
###### Lemma 2.1.

For any nonnegative integers , and , we have

 (2.11) h−jK∥q∥−j,K≂h−iK∥q∥−i,K∀ q∈Pℓ(K).
###### Proof.

It is sufficient to prove

 (2.12) ∥q∥0,K≂h−iK∥q∥−i,K∀ q∈Pℓ(K)

with . Applying the Poincaré-Friedrichs inequality (2.5) recursively, we get for any that

 (q,v)K≤∥q∥0,K∥v∥0,K≲hK∥q∥0,K|v|1,K≲⋯≲hiK∥q∥0,K|v|i,K.

Then it follows

 ∥q∥−i,K=supv∈Hi0(K)(q,v)K|v|i,K≲hiK∥q∥0,K,

which together with (2.10) yields (2.12). ∎

Recall the error estimates of the projection. For each and nonnegative integer , we have

 (2.13) ∥v−QKℓv∥0,K ≲hℓ+1K|v|ℓ+1,K∀ v∈Hℓ+1(K), (2.14) ∥v−QFℓv∥0,F ≲hℓ+1/2K|v|ℓ+1,K∀ v∈Hℓ+1(K).
###### Lemma 2.2.

We have for any that

 (2.15) ∥q∥0,K≲supw∈Pk(K;Rd)(divw,q)Kh−1K∥w∥0,K+|w|1,K.
###### Proof.

Due to (2.7), there exists such that

 divv=q|Ksh−1Ks∥v∥0,Ks+|v|1,Ks≲∥q∥0,Ks.

Let be the Brezzi-Douglas-Marini interpolation [9, 4], then

 IBDMKsv=v∀ v∈Pk(Ks;Rd),
 div(IBDMKsv)=QKsk−1divv=q|Ks,
 ∥v−IBDMKsv∥0,Ks≲hKs|v|1,Ks≲hKs∥q∥0,Ks.

It follows from the inverse inequality (2.10) and (2.13) that

 |IBDMKsv|1,Ks =|IBDMKsv−QKs0v|1,Ks≲h−1Ks∥IBDMKsv−QKs0v∥0,Ks ≲h−1Ks∥v−IBDMKsv∥0,Ks+h−1Ks∥v−QKs0v∥0,Ks ≲|v|1,Ks≲∥q∥0,Ks.

Noting that can be spontaneously extended to the domain , let such that . Thus

 (divw−q)|Ks=0,h−1Ks∥w∥0,Ks+|w|1,Ks≲∥q∥0,Ks.

Again due to being a polynomial,

 (divw−q)|Ks=0 imples divw=q on K.

And it follows from (2.9)

 h−1K∥w∥0,K+|w|1,K≲h−1Ks∥w∥0,Ks+|w|1,Ks≲∥q∥0,Ks≤∥q∥0,K.

Therefore we arrive at (2.15). ∎

## 3. Divergence-Free Nonconforming Virtual Elements

We will construct the divergence-free nonconforming virtual elements for Stokes problem in this section.

### 3.1. Virtual elements

For any , and satisfying , and for each , it follows from the integration by parts

 (ε(u),ε(v))K+(divv,p)K (3.1) = −(divε(u)+∇p,v)K+∑F∈F1(K)(ε(u)n+pn,v)F.

Following the ideas in [15, 22], suppose and temporarily. Inspired by the Green’s identity (3.1), we propose the following local degrees of freedom of the divergence-free nonconforming virtual elements for Stokes problem

 (3.2) (v,q)F ∀ q∈Pk−1(F;Rd) on each F∈F1(K), (3.3) (v,q)K ∀ q∈Pk−2(K;Rd)=G⊕k−2(K)⊕Gk−2(K).

Define the space of shape functions as

 Vk(K):={v∈H1(K;Rd): divv∈Pk−1(K), % there exists some s∈L2(K) such that divε(v)+∇s∈G⊕k−2(K), and (ε(v)n+sn)|F∈Pk−1(F;Rd)∀ F∈F1(K)}.

For any , since , by the direct sum decomposition (2.1) there exists such that . And then for each , thus

 Pk(K;Rd)⊆Vk(K).
###### Lemma 3.1.

The dimension of is same as the number of the degrees of freedom (3.2)-(3.3).

###### Proof.

Consider the local Stokes problem with the Neumann boundary condition

 (3.4) ⎧⎪⎨⎪⎩−div(ε(u))−∇p=f1in K,divu=f2in K,ε(u)n+pn=gFon each F∈F1(K),

where , , and . Employing the Green’s identity (3.1), we acquire

 (3.5) (ε(u),ε(v))K+(divv,p)K=(f1,v)K+∑F∈F1(K)(gF,v)F.

If taking in (3.5), we have the compatibility condition

 (3.6) (f1,q)K+∑F∈F1(K)(gF,q)F=0∀ q∈RM.

Given , , and satisfying the compatibility condition (3.6), due to (3.5), the weak formulation of the local problem (3.4) is to find and such that

 (3.7) ⎧⎨⎩(ε(u),ε(v))K+(divv,p)K=(f1,v)K+∑F∈F1(K)(gF,v)F,(divu,q)K=(f2,q)K,

for all and . According to the Babuška-Brezzi theory , the mixed formulation (3.7) is uniquely solvable.

On the other hand, if and there exist such that and , then , which implies , and thus is a constant. Hence the function in the definition of forms a one-dimensional vector space. As a result, we have

 dim(Vk(K)/RM)=dimPk−2(K;Rd)+∑F∈F1(K)dimPk−1(F;Rd)−dimRM.

Furthermore, if all the data , and vanish, then the solution of the local Stokes problem (3.4) belongs to . Therefore the dimension of is . ∎

###### Lemma 3.2.

The degrees of freedom (3.2)-(3.3) are unisolvent for the local virtual element space .

###### Proof.

Let and suppose all the degrees of freedom (3.2)-(3.3) vanish. We get from the integration by parts

 ∥divv∥20,K=−(∇divv,v)K+(divv,v⋅n)∂K=0.

Thus . Due to the definition of , there exists some such that , and for each . Then it follows from (3.1) that

 ∥ε(v)∥20,K+(divv,s)K=0,

which together with indicates . Applying the integration by parts,

 ∥∇v∥20,K=((∇v)n,v)∂K=0.

Therefore is constant, and thus , as required. ∎

###### Remark 3.3.

We have for , thus the virtual element is exactly the nonconforming element in .

### 3.2. Local projection

With the degrees of freedom (3.2)-(3.3), define a local operator as follows: given , let and be the solution of the local Stokes problem

 (3.8) (ε(ΠKkw),ε(v))K+(divv,PKw)K =(ε(w),ε(v))K∀ v∈Pk(K;Rd), (3.9) div(ΠKkw) =QKk−1(divw), (3.10) QK0(curlΠKkw) =QK0(curlw), (3.11) QK0(ΠKkw) =QK0w.

Similarly as (3.2) in , an equivalent formulation of the local Stokes problem (3.8)-(3.11) is

 ((ΠKkw,v))K+(divv,PKw)K =((w,v))K∀ v∈Pk(K;Rd), (div(ΠKkw),q)K =(divw,q)K∀ q∈Pk−1(K),

where

 ((w,v))K:=(ε(w),ε(v))K+QK0(curlw):QK0(curlv)+QK0w⋅QK0v

with symbols and being the inner products of the tensors and vectors respectively.

The inf-sup condition (2.15) indicates is a stable pair for Stokes problem, thus the local Stokes problem (3.8)-(3.11) is uniquely solvable. To simplify the notation, we will rewrite as . Apparently the projector can be computed using only the degrees of freedom (3.2)-(3.3). The unique solvability of the local Stokes problem (3.8)-(3.11) implies the operator is a projector, i.e.

 (3.12) ΠKq=q∀ q∈Pk(K;Rd).

It follows from (3.10)-(3.11), (2.13)-(2.14) and the Korn’s second inequality (2.6) that

 (3.13) ∥v∥0,K+hK|v|1,K+∑F∈F1(K)h1/2K∥v∥0,F≲hK∥ε(v)∥0,K∀ v∈ker(ΠK),

where . Due to (3.9), the local Stokes-based projector commutes with the divergence operator, i.e.

 (3.14) div(v−ΠKv)=0∀ v∈Vk(K).

By the Babuška-Brezzi theory , we get from the inf-sup condition (2.15) that

 ∥ε(ΠKw)∥0,K+∥PKw∥0,K ≲ supv∈Pk(K;Rd),q∈Pk−1(K)(ε(ΠKw),ε(v))K+(divv,PKw)K+(div(ΠKw),q)K∥ε(v)∥0,K+∥q∥0,K = supv∈Pk(K;Rd),q∈Pk−1(K)(ε(w),ε(v))K+(divw,q)K∥ε(v)∥0,K+∥q∥0,K,

which means the stability

 (3.15) ∥ε(ΠKw)∥0,K≲∥ε(w)∥0,K∀ w∈H1(K;Rd).

### 3.3. Norm equivalence

Given , let the stabilization

and the local bilinear form

 aKh(w,v):=(QKk−1ε(w),QKk−1ε(v))K+SK(w−ΠKw,v−ΠKv).

From (3.13) and (3.15), we have for any that

 (3.16)