# The lowest-order stabilizer free Weak Galerkin Finite Element Method

Recently, a new stabilizer free weak Galerkin method (SFWG) is proposed, which is easier to implement and more efficient. The main idea is that by letting j≥ j_0 for some j_0, where j is the degree of the polynomials used to compute the weak gradients, then the stabilizer term in the regular weak Galerkin method is no longer needed. Later on in <cit.>, the optimal of such j_0 for certain types of finite element spaces was given. In this paper, we propose a new efficient SFWG scheme using the lowest possible orders of piecewise polynomials for triangular meshes in 2 D with the optimal order of convergence.

## Authors

• 3 publications
• 5 publications
07/04/2019

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## 1 Introduction

In this paper, we are concerned with the development of an SFWG finite element method using the following Poisson equation

 (1) −Δu = f in Ω, (2) u = 0 on ∂Ω,

as the model problem, where is a polygonal domain in . The variational formulation of the Poisson problem (1)-(2) is to seek such that

 (3) (∇u,∇v) = (f,v),∀v∈H10(Ω).

The standard weak Glaerkin (WG) method for the problem (1)-(2) seeks weak Galerkin finite element solution such that

 (4) (∇wuh,∇wv)+s(uh,v) = (f,v),

for all with , where is the weak gradient operator and in (4) is a stabilizer term that ensures a sufficient weak continuity for the numerical approximation. The WG method has been developed and applied to different types of problems, including convection-diffusion equations [7, 6], Helmholtz equations [9, 12, 5], Stokes flow [11, 10], and biharmonic problems [8]. Recently, Al-Taweel and Wang in [2], proposed the lowest-order weak Galerkin finite element method for solving reaction-diffusion equations with singular perturbations in . One of major sources of the complexities of the WG methods and other discontinuous finite element methods is the stabilization term.

A stabilizer free weak Galerkin finite element method is proposed by Ye and Zhang in [3] as a new method for the solution of the Poisson equation on polytopal meshes in 2D or 3D, where elements are used. It is shown that there is a so that the SFWG method converges with optimal order of convergence for any . However, when is too large, the magnitude of the weak gradient can be extremely large, causing numerical instability. In [1], the optimal is given to improve the efficiency and avoid unnecessary numerical difficulties. In this setting, if elements are used for a triangular mesh, , where . In this paper, we propose a scheme using elements for triangular meshes with the optimal order of convergence, which is more efficient than using the regular WG method elements.

The goal of this paper is to develop the theoretical foundation for using the lowest-order SFWG scheme to solve the Poisson equation (1)-(2) on a triangle mesh in . The rest of this paper is organized as follows. In Section 2, the notations and finite element space are introduced. Section 3 is devoted to investigating the error equations and several other required inequalities. The main error estimate is studied in Section 4. In Section 5, we will derive the optimal order error estimates for the SFWG finite element method for solving the equations (1)-(2). Several numerical tests are presented in Section 6. Conclusions and some future research plans are summarized in Section 7.

## 2 Notations

In this section, we shall introduce some notations, and definitions.

Suppose is a quasi uniform triangular partition of . For every element , denote as its diameter and . Let be the set of all the edges in The weak Galerkin finite element space is defined as follows:

 (5) Vh={(v0,vb):v0∈P0(T),∀T∈Th, and vb∈P1(e),∀e∈Eh}.

In this instance, the component symbolizes the interior value of , and the component symbolizes the edge value of on each and , respectively. Let be the subspace of defined as:

 (6) V0h = {v:v∈Vh,vb=0 on ∂Ω}.

For each element , let be the -projection onto and let be the -projection onto . On each edge , denote by the -projection operator onto . Combining and , denote by the -projection operator onto

For any , the weak gradient is defined on as the unique polynomial satisfying

 (7) (∇wv,→q)T = −(v0,∇⋅→q)T+⟨vb,→q⋅→n⟩∂T,∀→q∈[P1(T)]2,

where

is the unit outward normal vector of

.

For simplicity, we adopt the following notations,

 (v,w)Th=∑T∈Th(v,w)T=∑T∈Th∫Tvwdx,
 ⟨v,w⟩∂Th=∑T∈Th⟨v,w⟩∂T=∑T∈Th∫∂Tvwdx.
###### Stabilizer free Weak Galerkin Algorithm 1

A numerical solution for (1)-(2) can be obtain by finding , such that the following equation holds

 (8) (∇wuh,∇wv)Th=(f,v0),∀v={v0,vb}∈V0h.

We define an energy norm on as:

 (9) |||v|||2=∑T∈Th(∇wv,∇wv)T.

An semi norm on is defined as:

 ∥v∥21,h=∑T∈Th(∥∇v0∥2T+h−1T∥v0−vb∥2∂T).
###### Remark 1

in the above definition is simply a placeholder, since .

Let

 (10) ^Vh={(v0,vb)|v0∈Pk(T),vb∈Pk(e),∀T∈Th,e∈∂T}.

The following lemma shows that and defined in (9) are equivalent on . (see [1]) Suppose that is convex with at most edges and satisfies the following regularity conditions: for all edges and of

 (11) |es|<α0|et|;

for any two adjacent edges the angle between them satisfies

 (12) θ0<θ<π−θ0,

where are independent of and . Let when each edge of is parallel to another edge of . Denote by the degree of weak gradient. When , then there exist two constants , such that for each , the following hold true

 C1∥v∥1,h,T≤|||v|||T≤C2∥v∥1,h,T,∀T∈Th,

where depend only on .

###### Remark 2

If all ’s are either triangles or parallelograms, then .

There exists so that

For any and , it follows from integration by part, trace inequality, and inverse inequality that

 (∇wv,→q)T = (∇v,→q)T+⟨v0−vb,→q⋅→n⟩∂T ≤ ∥∇v∥T∥→q∥T+Ch−12T∥v0−vb∥∂T∥→q∥T.

Letting yields the result.

The following lemma is one of the major results of this paper. For any , then,

 (13) ∥v∥1,h=∑T∈Th1h12T∥v0−vb∥∂T≤C|||∇wv|||.

Let . Since , it follows from Lemma 1 that

 (14) ∥v0−vb∥e2≤Ch12T1∥∇wv∥T1.

After a linear transformation we may assume

. Let be a unit tangent vector to such that . Denote . Let and let . Then

 |⟨v0−vb,→q⋅→n⟩e2| ≤ ∥v0−vb∥e2∥→q∥e2 = ∥v0−vb∥e2∥v0−vb∥e1.

Thus

 1√2∥v0−vb∥2e1 = ⟨v0−vb,(v0−vb+by)→t⋅→n1⟩e1 ≤ (∇wv,→q)T1+|⟨v0−vb,→q⋅→n⟩e2| ≤ ∥∇wv∥T1∥→q∥T1+∥v0−vb∥e2∥v0−vb∥e1.

Note that

 ∥→q∥2T1 = ∬T(a+bx+by)2dA = 12(a2+43ab+b22)=[ab]⎡⎣12131314⎤⎦[ab]=αTAα.

Note also that

Since is positive definite,

 (15) ∥→q∥2T1≤∥v0−vb∥2e1.

Thus

 1√2∥v0−vb∥2e1≤∥∇wv∥T1∥v0−vb∥e1+∥v0−vb∥e2∥v0−vb∥e1.

By a scaling argument and

 ∥v0−vb∥e1≤C(h12T1∥∇wv∥T1+∥v0−vb∥e2)≤Ch12T1∥∇wv∥T1.

Similarly, we can show that

 ∥v0−vb∥e3≤Ch12T1∥∇wv∥T1.

Thus we have

 ∥v0−vb∥∂T1≤Ch12T1∥∇wv∥T1.

Similarly, for any with at least one edge on ,

 ∥v0−vb∥∂T≤Ch12T∥∇wv∥T.

Let . Now, let’s look at and so that they share an edge . Without loss, we may assume that the vertices of are ,

 e1={(x,0)|0≤x≤1},

and the other edge of is , where . Denote . Since

 ∥v(2)0−vb∥e1≤∥v(2)0−v(1)0∥e1+∥v(1)0−vb∥e1

and

 ∥v(1)0−vb∥e1≤Ch12T1∥∇wv∥T1,

we need to show that

 ∥v(2)0−v(1)0∥e1≤C(∥∇wv∥T1+∥∇wv∥T2).

Let be unit tangents to , respectively; be linear functions such that . Let

 →q1=→q|T1=L3(x,y)(v(2)0−v(1)0)→t2

and

 →q2=→q|T2=L5(x,y)(v(2)0−v(1)0)→t4.

Then

 →qi⋅→n2i=0,→qi|e2i+1=0,i=1,2.

Scale if necessary so that

 −L3(0,0)→t2⋅→n1(1)=1=L5(0,0)→t4⋅→n1(2),

where is the unit outwards normal vector of .
Since ,

 L5(x,y)→t4⋅→n1(2)=~L5(x,y)=1−x−y.

Similarly

 L3(x,y)→t2⋅→n1(2)=^L3(x,y)=1−x+αy, for some α.

It follows from the shape regularity assumptions that the slope of , satisfies for some . Since ,

 (∇wv,→q)T1∪T2 = ⟨v0−vb,→q⋅→n⟩∂T1+⟨v0−vb,→q⋅→n⟩∂T2 = ⟨v(2)0−v(1)0,(v(2)0−v(1)0)~L3⟩e1.

Note that . Then

 (∇wv,→q)T1∪T2 = ⟨v(2)0−v(1)0,(v(2)0−v(1)0)~L3(x,y)⟩e1 = ∫10(v(2)0−v(1)0)2(1−x)dx = 12∥v(2)0−v(1)0∥2e1.

Thus

 ∥→q∥2T2=∬T2(1−x−y)2(v(2)0−v(1)0)2dA=112(v(2)0−v(1)0)2≤∥v(2)0−v(1)0∥2e1.

Similarly,

 ∥→q∥2T1=14(1+α+α23)(v(2)0−v(1)0)2≤14(1+1α0+13α20)∥v(2)0−v(1)0∥2e1.

Thus

 |(∇wv,→q)T1∪T2| ≤ ∥∇wv∥T1∥→q∥T1+∥∇wv∥T2∥→q∥T2 ≤ C∥∇wv∥T1∪T2⋅∥v(2)0−v(1)0∥e1.

Thus after a scaling we have

 ∥v(2)0−v(1)0∥e1≤Ch12T2∥∇wv∥T1∪T2.

Now let’s look at . Let

 →q=(v0−v(4)b+bx)→t5, where v0−v(4)b=a+by.

Then

 (∇wv,→q)T2=⟨v0−vb,→q⋅→n⟩e1+⟨v0−vb,→q⋅→n⟩e4,

which implies

 |(∇wv,→q)T2|≥|⟨v0−vb,→q⋅→n⟩e4|−|⟨v0−vb,→q⋅→n⟩e1|.

Thus

 |→t5⋅→n4|∥v0−vb∥2e4≤∥∇wv∥T2∥→q∥T2+∥v0−vb∥e1∥→q∥e1.

Since

 ∥→q∥2T2=12(a2+43ab+b22) and ∥v0−vb∥2e4=a2+ab+b23,
 ∥v0−vb∥2e4≥∥→q∥2T2.

Note that

 ∥→q∥2e1=∥→q∥2e4=∥v0−v2b∥e.

It follows from

 |→t5⋅→n4|∥v0−vb∥2e4≤∥∇wv∥T2∥→q∥T2+C∥v0−vb∥e1∥→q∥e1,

that

 ∥v0−vb∥e4≤C∥∇wv∥T2.

By a scaling argument,

 ∥v0−vb∥e4≤Ch12T2∥∇wv∥T2.

Similarly,

 ∥v0−vb∥e5≤Ch12T2∥∇wv∥T2.

Thus

 ∥v0−vb∥∂T2≤Ch12T∥∇wv∥T2∪T1≤Ch12T2(∥∇wv∥T1+∥∇wv∥T2).

Now let . So far we have shown that ,

 ∥v0−vb∥∂T≤Ch12T∥∇wv∥∪Ti≤Ch12T∑∥∇wv∥Ti,

where and shares at least one edge with . This process can be continued, until we have visited every and have

 ∥v0−vb∥∂T≤Ch12T∥∇wv∥∪Ti≤Ch12T∑∥∇wv∥Ti,

where shares at least edge with . (13) follows from this immediately.

Combining Lemmas 2 and 2, we have the following theorem. There exists such that

## 3 Error equation

In this section, we derive the error estimate Algorithm 1.

For any function , the following trace inequality holds true:

 (16) ∥ψ∥2e≤C(h−1T∥ψ∥2T+hT∥∇ψ∥2T).

(Inverse Inequality) There exists a constants such that for any piecewise polynomial ,

 (17) ||∇ψ||T≤Ch−1T||ψ||T,∀T∈Th.

Let be the solution of the problem and be a finite element partition of satisfying the shape regularity assumptions. Then, the projections satisfies

 (18) ∑T∈Th(∥u−Q0u∥2T+h2T∥∇(u−Q0u)∥2T)≤Ch2∥u∥21, (19) ∑T∈Th(∥∇u−Qh∇u∥2T+h2T∥∇u−Qh∇u)∥21,T)≤Ch2(1+i)∥u∥22+i,i=0,1.

Let . Then for each element , we have

 (20) Qh(∇ϕ)=∇wQhϕ.

By definition (7) and integration by parts, for each we have

 (Qh(∇ϕ),→q)T = −(ϕ,∇⋅→q)T+⟨ϕ,→q⋅→n⟩∂T = −(Q0ϕ,∇⋅→q)T+⟨Qbϕ,→q⋅→n⟩∂T = (∇wQhϕ,→q)T.

which implies (20). Let . Then for all , we have

 (21) (∇ϕ,∇v0)T=(∇w(Qhϕ),∇wv)T + ⟨(Qh(∇ϕ)⋅→n,v0−vb⟩∂T.

Let and . By Lemma 3 .

 (22) (→q,∇v0)T = −(∇⋅→q,v0)T+⟨→q⋅→n,v0⟩∂T, (23) (→q,∇wv)T = −(∇⋅→q,v0)T+⟨→q⋅→n,vb⟩∂T,

implies

 (→q,∇v0)T = (→q,∇wv)T+⟨→q⋅→n,v0−vb⟩∂T = (∇wP,∇wv)T+⟨→q⋅→n,v0−vb⟩∂T,

which completes the proof. Let . Then for any , we have

 (24) (∇weh,∇wv)Th=ℓ(u,v),

where is defined as follows,

 ℓ(u,v)=∑T∈Th⟨(∇u−Qh∇u)⋅→n,v0−vb⟩∂T.

Testing the equation (1) by and using the fact that , we arrive at