# A note on the optimal degree of the weak gradient of the stabilizer free weak Galerkin finite element method

Recently, a new stabilizer free weak Galerkin method (SFWG) is proposed, which is easier to implement. The idea is to raise the degree of polynomials j for computing weak gradient. It is shown that if j>=j0 for some j0, then SFWG achieves the optimal rate of convergence. However, large j will cause some numerical difficulties. To improve the efficiency of SFWG and avoid numerical locking, in this note, we provide the optimal j0 with rigorous mathematical proof.

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## 1 Introduction

In this note, similar to [1], we will consider 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).

A stabilizer free weak Galerkin finite element method is proposed by Ye and Zhang in [1] as a new method for the solution of Poisson equation on polytopal meshes in 2D or 3D. Let be the degree of polynomials for the calculation of weak gradient. It has been proved in [1] 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 this note, we provide the optimal to improve the efficiency and avoid unnecessary numerical difficulties, which has mathematical and practical interests.

## 2 Weak Galerkin Finite Element Schemes

Let be a partition of the domain consisting of polygons in 2D. Suppose that is shape regular in the sense defined by (2.7)-(2.8) and each is convex. Let be the set of all edges in , and let be the set of all interior edges. For each element , denote by the diameter of , and the mesh size of .

On , let be the space of all polynomials with degree no greater than . Let be the weak Galerkin finite element space associated with defined as follows:

 (4) Vh = {v={v0,vb}:v0|T∈Pk(T),vb|e∈Pk(e),e∈∂T,T∈Th},

where is a given integer. 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:

 (5) V0h = {v:v∈Vh,vb=0 on ∂Ω}.

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

 (6) (∇wv,q)T = −(v0,∇⋅q)T+⟨vb,q⋅n⟩∂T,∀q∈[Pj(T)]2,j≥k.

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.

For any and in , we define the bilinear forms as follows:

 (7) A(v,w) =∑T∈Th(∇wv,∇ww)T.
###### Stabilizer free Weak Galerkin Algorithm 1

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

 (8) A(uh,v)=(f,v0),∀v={v0,vb}∈V0h.

where is defined in (7).

Accordingly, for any , we define an energy norm on as:

 (9) |||v|||2=∑T∈Th(∇wv,∇wv)T=∑T∈Th∥∇wv∥2T.

An norm on is defined as:

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

The following well known lemma and corollary will be needed in proving our main result. Suppose that are convex regions in such that can be obtain by scaling with a factor . Then for any ,

 ∥p∥D1≤∥p∥D2≤Crn∥p∥D1,

where depends only on .

###### Corollary 2.1

Suppose that satisfy the conditions of Lemma 2 and . Then

 ∥p∥D1≤∥p∥D≤Crn∥p∥D1,∀p∈Pn(D),

where depends only on .

The following Theorem shows that for certain , whenever , is equivalent to the defined in (9), which is crucial in establishing the feasibility of SFWG. Later on, we will show that is optimal. (cf.[1], Lemma 3.1) Suppose that is convex with at most edges and satisfies the following regularity conditions: for all edges and of

 (10) |es|<α0|et|;

for any two adjacent edges the angle between them satisfies

 (11) θ0<θ<π−θ0,

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

where depend only on . For simplicity, from now on, all constant are independent of and , unless otherwise mentioned. They may depend on . Suppose . We know that

 (12) (∇wv,→q)T=(∇v,→q)T+⟨vb−v0,→q⋅→n⟩∂T∀→q∈[Pj(T)]2.

Suppose , and we want to construct a , where , so that

 (13) (→q,p)T = 0,∀p∈[Pk−1(T)]2, (14) →q⋅→ni = 0 on ei,i=1,…,m−1, (15) ⟨vb−v0−→q⋅n0,p⟩e0 = 0,p∈Pk(e0),

where is the unit outer normal to . Without loss of generality, we may assume that , and one of the vertices of is . Let be such that . Let

 (16) →q=ℓ2…ℓm−1Q1→t=LQ1→t=Q→t,

where is a unit tangent vector to and . It is easy to see that . We want to find so that

 (17) ⟨vb−v0−ℓ2…ℓmQ1(→t1⋅→n0),p⟩e0 = 0,∀p∈Pj0−m+2(e0), (18) (ℓ2…ℓmQ1,p)T = 0,∀p∈Pk−1(T).

Let or . It is easy to see that if satisfies (17)-(18), then satisfies (13)-(15). In addition, (17)-(18) has equations and the same number of unknowns. To show (17)-(18) has a unique solution we set . Setting in (17) yields

 ⟨ℓ2…ℓm−1Q1,Q1⟩e0=0.

Since inside , . Thus , where . Similarly

 (yℓ2…ℓm−1Q2,p)T=0∀p∈Pk−1(T),

implies and thus . Note that for any , since , there exists at least one such in . Note also that when every edge of is parallel to another one, we can lower the number of in (16) by one. Thus for such a , we have . Next, we want to show that the unique solution of (13)-(15) satisfies

 (19) ∥→q∥T≤γ0∥→q⋅→n0∥e0,

for some . Note that

 (→q⋅→n0)2=Q2sin2θ=→q⋅→qsin2θ≥→q⋅→qsin2θ0,

where is the angle between and . Thus, (19) is equivalent to

 (20) ∥Q∥T≤γ0∥Q∥e0,

for some and we may assume

 e1={(0,y)|0≤y≤1},

because of the assumption (10). Now let be the triangle spanned by and and label the third edge of as . Let be the triangle with vertices and . Label the edge of connecting and as . For , let

 di = dist(^Tr,Li), Di = maxp∈Tdist(p,Li),

where is the line containing . Then

 di ≥ dist(^et,et)=√2/4, (21) Di ≤ (|e0|+|e1|+⋯+|em−1|)/2 ≤ mα02.

Thus ,

 (22) ℓi(p)≥√22mα0=ε,i=2,…,m−1.

Now we will prove (20) in steps:

 I:β1∥Q1∥e0 ≤ ∥Q∥e0, II:β2∥Q∥Tr ≤ ∥Q1∥e0, III:β3∥Q∥T ≤ ∥Q∥Tr.
1. Let . Then similar to (22), . Thus

 εm−2∥Q1∥^e≤∥Q∥^e≤∥Q∥e0.

It follows from Corollary 2.1 that

 (23) ∥Q∥e0≥εm−2∥Q1∥^e≥Cεm−2∥Q1∥e0=β1∥Q1∥e0.
2. It is easy to see that

 ∥Q∥2Tr = ∫10∫1−y0L2Q21dxdy=∫10∫1−y0L2(Q1−Q1(x,0)+Q1(x,0))2dxdy ≤ 2∫10∫1−y0[L2(Q1−Q1(x,0))2+L2(Q1(x,0))2]dxdy.

It follows from (22) and Corollary 2.1 that

 (24) ∥LQ1(x,0)∥2Tr = ∫10∫1−y0L2(Q1(x,0))2dxdy ≤ ∫10∫10(Q1(x,0))2dxdy = ∥Q1∥2e0

Note that

 Q1−Q1(x,0)=y¯Q,¯Q∈Pk−1,

and

 (25) ∫T[LQ1(x,0)+Ly¯Q]PdA=0,∀p∈Pk−1.

Since ,

 (26) ∫TLy¯Q2dA≥ε4∫^Tr¯Q2dA=δ3∥¯Q∥2^Tr.

Since are equivalent norms on ,

 (27) δ3∥¯Q∥^Tr≥β3∥¯Q∥Tr.

Let be the circumscribed isosceles-right triangle obtained by scaling . It is easy to see that

 (28) diam(Tce0)≤2mα0.

Since , it follows from (26), (27), and Corollary 2.1 that

 ∫TLy¯Q2dA≥β3∥¯Q−1∥2Tr≥δ4∥¯Q∥2T,

where depend on (28) and . Letting in (25) yields

 ∫TLy¯Q2dA=−∫TLQ1(x,0)¯QdA≤∥Q1(x,0)∥T∥¯Q∥T,

and thus

 ∥¯Q∥T≤1δ4∥Q1(x,0)∥T.

Then

 ∥¯Q∥2T≤δ5∫N0∫M0Q21(x,0)dxdy=CN∫M0Q21(x,0)dx=C∥Q1∥2[0,M],

where . It follows from Corollary 2.1 and (26)

 (29) ∥Ly¯Q∥^Tr≤∥¯Q∥T≤C∥Q1∥e0.

Combining (29) and (24) yields

 (30) ∥Q∥Tr≤C∥Q1∥e0.
3. Applying Corollary 2.1 again yields

 (31) C∥Q∥T≤∥Q∥Tr.

Now we have completed the -step argument.
By a scaling argument, we have

 (32) ∥Q∥T≤γ0h1/2∥Q∥e0.

Plugging into (12) yields

 (33) ∥Q∥T∥∇wv∥T ≥ (∇wv,→q→n)^T=⟨vb−v0,→q→n⋅→n⟩e0 = ∥vb−v0∥2e0 = ∥vb−v0∥e0∥Q∥e0.

It follows from (32) that

 ∥∇wv∥T≥Ch1/2∥vb−v0∥e0.

Similarly,

 ∥vb−v0∥ei≤Ch1/2T∥∇wv∥T,i=1,…,m−1.

Thus

 ∥vb−v0∥∂T≤Ch1/2T∥∇wv∥T.

By letting in (12), we arrive at

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

It follows from the trace inequality and the inverse inequality that

 ∥∇wv∥2T ≤ ∥∇v0∥T∥∇wv∥T+∥v0−vb∥∂T∥∇wv∥∂T ≤ ∥∇v0∥T∥∇wv∥T+Ch1/2T∥v0−vb∥∂T∥∇wv∥T.

Thus

 ∥∇wv∥T ≤ C(∥∇v0∥T+h1/2T∥v0−vb∥∂T).

To show that is the lowest possible degree for the weak gradient, we give the following examples.

###### Example 2.1

Suppose and is partitioned as shown in Figure 1.

Let . To see if (8) has a unique solution in , we set and . Then . By definition,

 0=(∇wuh,→q)Th=−(uh,∇→q)Th+⟨ub,→q⋅→n⟩∂Th,∀→q∈[P1(Th)]2.

Note that the degree of freedom of

is while the degree of freedom of is . Thus, there exist so that . Thus Algorithm will not work.

###### Example 2.2

Let and is partitioned as shown in Figure 2.

Let . Note that the degree of freedom of is while the degree of freedom of is . A similar argument as in example 2.1 shows that Algorithm will not work.

For the completeness, we list the following from [1]. Let be the weak Galerkin finite element solution of (8). Assume that the exact solution and -regularity holds true. Then, there exists a constant such that

 (34) |||u−uh||| ≤ Chk|u|k+1, (35) ∥u−u0∥ ≤ Chk+1|u|k+1.

## 3 Numerical Experiments

We are devoting this section to verify our theoretical results in previous sections by two numerical examples. The domain in all examples is . We will implement the SFWG finite element method (8) on triangular meshes. A uniform triangulation of the domain is used. By refining each triangle for , we obtain a sequence of partitions. The first two levels of grids are shown in Figure 3. We apply the SF-WG finite element method with finite element space to find SFWG solution in the computation.

###### Example 3.1

In this example, we solve the Poisson problem (1)-(2) posed on the unit square , and the analytic solution is

 (36) u(x,y)=sin(πx)sin(πy).

The boundary conditions and the source term are computed accordingly. Table 1 lists errors and convergence rates in -norm and -norm.

###### Example 3.2

In this example, we consider the problem with boundary condition and the exact solution is

 (37) u(x,y)=x(1−x)y(1−y).

The results obtained in Table 2 show the errors of SFWG scheme and the convergence rates in the norm and norm. The simulations are conducted on triangular meshes and polynomials of order . The SFWG scheme with element has optimal convergence rate of in -norm and in -norm.

Overall, the numerical experiment’s results shown in the formentioned examples match the theoretical study part in the previous sections.

## References

• [1] X. Ye, S. Zhang, On stabilizer-free weak Galerkin finite element method on polytopal meshes, arXiv:1906.06634v2, 2019.