Numerical investigation on weak Galerkin finite elements

1 Introduction

For simplicity, we demonstrate the idea by using the second order elliptic problem that seeks an unknown function $u$ satisfying

(1.1)		$- \nabla \cdot (\nabla u)$	$=$	$f in Ω,$
(1.2)		$u$	$=$	$g on \partial Ω,$

where $Ω$ is a polytopal domain in $R^{2}$ .

The weak form of the problem (1.1)-(1.2) is to find $u \in H^{1} (Ω)$ such that $u = g$ on $\partial Ω$ and satisfies

(1.3)

(\nabla u, \nabla v) = (f, v) \forall v \in H_{0}^{1} (Ω) .

The weak Galerkin finite element method is an effective and flexible numerical technique for solving partial differential equations. It is a natural extension of the standard Galerkin finite element method where classical derivatives were substituted by weakly defined derivatives on functions with discontinuity. The WG method was first introduced in [17, 18] and then has been applied to solve various PDEs such as second order elliptic equations, biharmonic equations, Stokes equations, Navier-Stokes equations, Brinkman equations, parabolic equations, Helmholtz equation, convection dominant problems, hyperbolic equations, and Maxwell’s equations [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19].

The main idea of weak Galerkin finite element methods is the use of weak functions and their corresponding weak derivatives. For the second order elliptic equation, weak functions have the form of $v = {v_{0}, v_{b}}$ with $v = v_{0}$ inside of each element and $v = v_{b}$ on the boundary of the element. Both $v_{0}$ and $v_{b}$ can be approximated by polynomials in $P_{ℓ} (T)$ and $P_{s} (e)$ respectively, where $T$ stands for an element and $e$ the edge or face of $T$ , $ℓ$ and $s$ are non-negative integers with possibly different values. Weak derivatives are defined for weak functions in the sense of distributions. Denote by $G_{m} (T)$

the vector space for weak gradient. Typical choices for

G_{m} (T)

are

[P_{m} (T)]^{d}

R T_{m} (T)

. Various combination of

(P_{ℓ} (T), P_{s} (e), G_{m} (T))

leads to different weak Galerkin methods tailored for specific partial differential equations.

Weak Galerkin finite element methods have two forms for the problem (1.1)-(1.2). One is its standard formulation [11, 17]: find $u_{h} \in V_{h}$ such that $u_{h} = Q_{b} g$ on $\partial Ω$ and satisfies

(1.4)

(\nabla_{w} u_{h}, \nabla_{w} v) + s (u_{h}, v) = (f, v) \forall v \in V_{h}^{0},

where $s (\cdot, \cdot)$ is a parameter independent stabilizer. Another one is WG stabilizer free formulation [1, 20, 21]: find $u_{h} \in V_{h}$ such that $u_{h} = Q_{b} g$ on $\partial Ω$ and satisfies

(1.5)

(\nabla_{w} u_{h}, \nabla_{w} v) = (f, v) \forall v \in V_{h}^{0} .

Removing stabilizers simplifies the formulations and reduces programming complexity. A stabilizer free WG method can be obtained by raising the degree of polynomial $m$ for approximating weak gradient in the WG element $(P_{ℓ} (T), P_{s} (e), G_{m} (T))$ .

The purpose of this paper is to investigate the performance of different WG elements computationally in the weak Galerkin finite element methods with or without stabilizers. Like a periodic table, we provide 31 tables that are informative and clearly demonstrate special properties of each WG element. We don’t have all the theoretical answers for many interesting phenomena shown in the tables and we leave them for interesting readers.

While preparing this manuscript, three papers are in the process to answer some questions from the numerical results in the tables. We are close to prove theoretically that the WG element $(P_{k} (T), P_{k} (e), [P_{k + 1}]^{2})$ has two orders of supercloseness in both energy norm and $L^{2}$ norm on rectangular meshes, shown in Table 3.3. It is proved in [2] that the WG element $(P_{k} (T), P_{k + 1} (e), [P_{k + 1}]^{2})$ has two orders of supercloseness in both energy norm and $L^{2}$ norm, on general triangular meshes in Table 6.3. Due to the bad behavior of the WG element $(P_{k} (T), P_{k - 1} (e), [P_{k + 1}]^{2})$ shown in Table 7.3 and 8.3, a new definition of the weak gradient is introduced in [22] so that the element can still converge in optimal order on general polytopal meshes.

The WG methods are designed for using discontinuous approximations on general polytopal meshes. Due to limited space, we only consider the finite element partitions including rectangles and triangles.

2 Weak Galerkin Finite Element formulations

Let $T_{h}$ be a partition of the domain $Ω$ consisting of rectangles or triangles. Denote by $E_{h}$ the set of all edges in $T_{h}$ , and let $E_{h}^{0} = E_{h} ∖ \partial Ω$ be the set of all interior edges or flat faces. For every element $T \in T_{h}$ , we denote by $h_{T}$ its diameter and mesh size $h = {max}_{T \in T_{h}} h_{T}$ for $T_{h}$ . Let $P_{k} (T)$ consist all the polynomials defined on $T$ of degree less or equal to $k$ .

Definition 1

For $T \in T_{h}$ and $ℓ, s \geq 0$ , define a local WG element $W_{ℓ, s} (T)$ as,

(2.1)

W_{ℓ, s} (T) = {v = {v_{0}, v_{b}} : v_{0} |_{T} \in P_{ℓ} (T), v_{b} |_{e} \in P_{s} (e), e \subset \partial T} .

Definition 2

For any $v = {v_{0}, v_{b}} \in W_{ℓ, s} (T)$ , a weak gradient $\nabla_{w} v \in G_{m} (T)$ is defined as a unique solution of the following equation

(2.2)

(\nabla_{w} v, q)_{T} = - (v_{0}, \nabla \cdot q)_{T} + ⟨ v_{b}, q \cdot n ⟩_{\partial T} \forall q \in G_{m} (T) .

A typical choice of $G_{m} (T)$ is $[P_{m} (T)]^{d}$ , or $R T_{m} (T)$ . Different combinations of $(ℓ, s, m)$ associated with a WG element $W_{ℓ, s} (T) / G_{m} (T)$ leads to different weak Galerkin finite element formulations. The weak gradient $\nabla_{w}$ defined in (2.2) is an approximation of $\nabla$ that is computed on each element $T$ .

Remark 1

Please note that the space $G_{m} (T)$

is used to calculated weak gradient and does not introduce additional degrees of freedom to the resulting linear system.

Definition 3

Define a WG finite element space $V_{h}$ associated with $T_{h}$ as follows

(2.3)

V_{h} = {v = {v_{0}, v_{b}} : v |_{T} \in W_{ℓ, s} (T), \forall T \in T_{h}} .

We would like to emphasize that any function $v \in V_{h}$ has a single value $v_{b}$ on each edge $e \in E_{h}$ . The subspace of $V_{h}$ consisting of functions with vanishing boundary value is denoted as $V_{h}^{0}$ .

Let $Q_{0}$ and $Q_{b}$ be the two element-wise defined $L^{2}$ projections onto $P_{ℓ} (T)$ and $P_{s} (e)$ on each $T \in T_{h}$ , respectively. Define $Q_{h} u = {Q_{0} u, Q_{b} u} \in V_{h}$ . Let $Q_{h}$ be the element-wise defined $L^{2}$ projection onto $G_{m} (T)$ on each element $T \in T_{h}$ .

For simplicity, we adopt the following notations,

	$(v, w)_{T_{h}}$	$=$	$\sum T \in T_{h} (v, w)_{T} = \sum T \in T_{h} \int_{T} v w d x,$
	$⟨ v, w ⟩_{\partial T_{h}}$	$=$	$\sum T \in T_{h} ⟨ v, w ⟩_{\partial T} = \sum T \in T_{h} \int_{\partial T} v w d s .$

Weak Galerkin Algorithm 1

A numerical approximation for (1.1)-(1.2) can be obtained by seeking $u_{h} = {u_{0}, u_{b}} \in V_{h}$ satisfying $u_{b} = Q_{b} g$ on $\partial Ω$ and the following equation:

(2.4)

(\nabla_{w} u_{h}, \nabla_{w} v)_{T_{h}} + s (u_{h}, v) = (f, v_{0}) \forall v = {v_{0}, v_{b}} \in V_{h}^{0},

where the stabilizer $s (\cdot, \cdot)$ is defined as

(2.5)

s (u_{h}, v) = \sum T \in T_{h} h_{T}^{j} ⟨ Q_{b} u_{0} - u_{b}, Q_{b} v_{0} - v_{b} ⟩_{\partial T} .

Let $j = \infty$ in (2.5), we mean $s (u_{h}, v) = 0$ , i.e., we have the following stabilizer-free WG formulation,

(2.6)

(\nabla_{w} u_{h}, \nabla_{w} v)_{T_{h}} = (f, v_{0}) \forall v = {v_{0}, v_{b}} \in V_{h}^{0} .

In the following sections, we will conduct extensive numerical tests to study the performance of different WG elements and record the results in 31 tables. In all the tables below, $j = \infty$ refers to the stabilizer free WG formulation (2.6), where $j$ is defined in (2.5).

3 The WG elements with $ℓ = s$ on rectangular mesh

Next we will study convergence rate for the WG element $(P_{ℓ} (T), P_{s} (e), G_{m} (T))$ with $ℓ = s$ on rectangular meshes. The rectangular meshes used in the computation are illustrated in Figure 3.1.

Figure 3.1: The first three level rectangular grids.

Table 3.1 demonstrates the convergence rates for $(P_{k} (T), P_{k} (e), [P_{k - 1} (T)]^{2})$ with a stabilizer of different $j$ defined in (2.5) on rectangular mesh.

element	$P_{k} (T)$	$P_{k} (e)$	$[P_{k - 1} (T)]^{2}$	$j$	$r_{1}$	$r_{2}$	Proved
3.1.1				$- 1$	$1$	$2$	Yes
3.1.2				$0$	$0.5$	$1$	No
3.1.3	$P_{1} (T)$	$P_{1} (e)$	$[P_{0} (T)]^{2}$	$1$	$0$	$0$	No
3.1.4				$\infty$	$- \infty$	$- \infty$	No
3.1.5				$- 1$	$2$	$3$	Yes
3.1.6				$0$	$1.5$	$2$	No
3.1.7	$P_{2} (T)$	$P_{2} (e)$	$[P_{1} (T)]^{2}$	$1$	$1$	$1$	No
3.1.8				$\infty$	$- \infty$	$- \infty$	No
3.1.9				$- 1$	$3$	$4$	Yes
3.1.10				$0$	$2.5$	$3$	No
3.1.11	$P_{3} (T)$	$P_{3} (e)$	$[P_{2} (T)]^{2}$	$1$	$2$	$2$	No
3.1.12				$\infty$	$- \infty$	$- \infty$	No

Table 3.1: Element

(P_{k} (T), P_{k} (e), [P_{k - 1} (T)]^{2})

on rectangular mesh,

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

Table 3.2 demonstrates the convergence rates for $(P_{k} (T), P_{k} (e), [P_{k} (T)]^{2})$ with a stabilizer of different $j$ on rectangular mesh.

element	$P_{k} (T)$	$P_{k} (e)$	$[P_{k} (T)]^{2}$	$j$	$r_{1}$	$r_{2}$	Proved
3.2.1				$- 1$	$0$	$0$	No
3.2.2				$0$	$0.5$	$1$	No
3.2.3	$P_{0} (T)$	$P_{0} (e)$	$[P_{0} (T)]^{2}$	$1$	$0$	$0$	No
3.2.4				$\infty$	$- \infty$	$- \infty$	No
3.2.5				$- 1$	$1$	$2$	No
3.2.6				$0$	$1.5$	$2$	No
3.2.7	$P_{1} (T)$	$P_{1} (e)$	$[P_{1} (T)]^{2}$	$1$	$1$	$1$	No
3.2.8				$\infty$	$- \infty$	$- \infty$	No
3.2.9				$- 1$	$2$	$3$	No
3.2.10				$0$	$2.5$	$3$	No
3.2.11	$P_{2} (T)$	$P_{2} (e)$	$[P_{2} (T)]^{2}$	$1$	$2$	$2$	No
3.2.12				$\infty$	$- \infty$	$- \infty$	No

Table 3.2: Element

(P_{k} (T), P_{k} (e), [P_{k} (T)]^{2})

on rectangular mesh,

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

Remark 2

Theorem 4.9 in [16] guarantees the optimal convergence rate of the WG element 3.2.5 in the $| | | \cdot | | |$ norm. However the optimal convergence rate in the $L^{2}$ norm is not proved in [16]. Therefore, we still mark $p r o v e d = N o$ in the case 3.2.5.

Table 3.3 demonstrates the convergence rates for $(P_{k} (T), P_{k} (e), [P_{k + 1} (T)]^{2})$ with a stabilizer of different $j$ on rectangular mesh.

element	$P_{k} (T)$	$P_{k} (e)$	$[P_{k + 1} (T)]^{2}$	$j$	$r_{1}$	$r_{2}$	Proved
3.3.1				$- 1$	$0$	$0$	No
3.3.2				$0$	$1$	$1$	No
3.3.3	$P_{0} (T)$	$P_{0} (e)$	$[P_{1} (T)]^{2}$	$1$	$2$	$2$	No
3.3.4				$\infty$	$2$	$2$	No
3.3.5				$- 1$	$1$	$2$	No
3.3.6				$0$	$2$	$3$	No
3.3.7	$P_{1} (T)$	$P_{1} (e)$	$[P_{2} (T)]^{2}$	$1$	$3$	$4$	No
3.3.8				$\infty$	$3$	$4$	Yes
3.3.9				$- 1$	$2$	$3$	No
3.3.10				$0$	$3$	$4$	No
3.3.11	$P_{2} (T)$	$P_{2} (e)$	$[P_{3} (T)]^{2}$	$1$	$4$	$5$	No
3.3.12				$\infty$	$4$	$5$	Yes

Table 3.3: Element

(P_{k} (T), P_{k} (e), [P_{k + 1} (T)]^{2})

on rectangular mesh,

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

Table 3.4 demonstrates the convergence rates for $(P_{k} (T), P_{k} (e), [P_{k + 2} (T)]^{2})$ with a stabilizer of different $j$ on rectangular mesh.

element	$P_{k} (T)$	$P_{k} (e)$	$[P_{k + 2} (T)]^{2}$	$j$	$r_{1}$	$r_{2}$	Proved
3.4.1				$- 1$	$0$	$0$	No
3.4.2				$0$	$0$	$0$	No
3.4.3	$P_{0} (T)$	$P_{0} (e)$	$[P_{2} (T)]^{2}$	$1$	$0$	$0$	No
3.4.4				$\infty$	$0$	$0$	No
3.4.5				$- 1$	$1$	$2$	No
3.4.6				$0$	$1$	$2$	No
3.4.7	$P_{1} (T)$	$P_{1} (e)$	$[P_{3} (T)]^{2}$	$1$	$1$	$2$	No
3.4.8				$\infty$	$1$	$2$	Yes
3.4.9				$- 1$	$2$	$3$	No
3.4.10				$0$	$2$	$3$	No
3.4.11	$P_{2} (T)$	$P_{2} (e)$	$[P_{4} (T)]^{2}$	$1$	$2$	$3$	No
3.4.12				$\infty$	$2$	$3$	Yes

Table 3.4: Element

(P_{k} (T), P_{k} (e), [P_{k + 2} (T)]^{2})

on rectangular mesh,

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

Remark 3

For the $P_{k} (T) - P_{k} (e)$ element, Tables 3.1-3.4 demonstrate that the performance of the WG solutions are getting better when the degree of the polynomials for weak gradient is increasing from $k - 1$ to $k + 1$ . Specially the WG element $(P_{k} (T), P_{k} (e), [P_{k + 1} (T)]^{2})$ shows order two supercloseness in Table 3.3. However, for the element $(P_{k} (T), P_{k} (e), [P_{k + 2} (T)]^{2}$ , the numerical tests in Table 3.4 show the convergence rate of the WG solution decreasing. Remember that increasing $m$ in $[P_{m} (T)]^{2}$ for weak gradient does not introduce additional degrees of freedom for the resulting linear systems.

Table 3.5 demonstrates the convergence rates for $(P_{k} (T), P_{k} (e), R T_{k} (T))$ with a stabilizer of different $j$ on rectangular mesh.

element	$P_{k} (T)$	$P_{k} (e)$	$R T_{k} (T)$	$j$	$r_{1}$	$r_{2}$	Proved
3.5.1				$- 1$	$0$	$0$	No
3.5.2				0	$1$	$1$	No
3.5.3	$P_{0} (T)$	$P_{0} (e)$	$R T_{0} (T)$	1	$2$	$2$	No
3.5.4				$\infty$	$2$	$2$	No
3.5.5				$- 1$	$1$	$2$	No
3.5.6				0	$1.5$	$2$	No
3.5.7	$P_{1} (T)$	$P_{1} (e)$	$R T_{1} (T)$	1	$1$	$1$	No
3.5.8				$\infty$	$1$	$1$	No
3.5.9				$- 1$	$2$	$3$	No
3.5.10				0	$2.5$	$3$	No
3.5.11	$P_{2} (T)$	$P_{2} (e)$	$R T_{2} (T)$	1	$2$	$2$	No
3.5.12				$\infty$	$2$	$2$	No

Table 3.5: Element

(P_{k} (T), P_{k} (e), R T_{k} (T))

on rectangular mesh,

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

4 The WG elements for $ℓ = s$ on triangular mesh

The triangular meshes used in the computation are displayed in Figure 4.1.

Figure 4.1: The first three level triangular meshes.

Table 4.1 demonstrates the convergence rates for $(P_{k} (T), P_{k} (e), [P_{k - 1} (T)]^{2})$ with a stabilizer of different $j$ on triangular mesh.

element	$P_{k} (T)$	$P_{k} (e)$	$[P_{k - 1} (T)]^{2}$	$j$	$r_{1}$	$r_{2}$	Proved
4.1.1				$- 1$	$1$	$2$	Yes
4.1.2				0	$0.5$	$1$	No
4.1.3	$P_{1} (T)$	$P_{1} (e)$	$[P_{0} (T)]^{2}$	1	$0$	$0$	No
4.1.4				$\infty$	$- \infty$	$- \infty$	No
4.1.5				$- 1$	$2$	$3$	Yes
4.1.6				0	$1.5$	$2$	No
4.1.7	$P_{2} (T)$	$P_{2} (e)$	$[P_{1} (T)]^{2}$	1	$1$	$1$	No
4.1.8				$\infty$	$- \infty$	$- \infty$	No
4.1.9				$- 1$	$3$	$4$	Yes
4.1.10				0	$2.5$	$3$	No
4.1.11	$P_{3} (T)$	$P_{3} (e)$	$[P_{2} (T)]^{2}$	1	$2$	$2$	No
4.1.12				$\infty$	$- \infty$	$- \infty$	No

Table 4.1: Element

(P_{k} (T), P_{k} (e), [P_{k - 1} (T)]^{2})

on triangular mesh,

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

Table 4.2 demonstrates the convergence rates for $(P_{k} (T), P_{k} (e), [P_{k} (T)]^{2})$ with a stabilizer of different $j$ on triangular mesh.

element	$P_{k} (T)$	$P_{k} (e)$	$[P_{k} (T)]^{2}$	$j$	$r_{1}$	$r_{2}$	Proved
4.2.1				$- 1$	$0$	$0$	No
4.2.2				0	$0.5$	$1$	No
4.2.3	$P_{0} (T)$	$P_{0} (e)$	$[P_{0} (T)]^{2}$	1	$0$	$0$	No
4.2.4				$\infty$	$- \infty$	$- \infty$	No
4.2.5				$- 1$	$1$	$2$	No
4.2.6				0	$1.5$	$2$	No
4.2.7	$P_{1} (T)$	$P_{1} (e)$	$[P_{1} (T)]^{2}$	1	$1$	$1$	No
4.2.8				$\infty$	$- \infty$	$- \infty$	No
4.2.9				$- 1$	$2$	$3$	No
4.2.10				0	$2.5$	$3$	No
4.2.11	$P_{2} (T)$	$P_{2} (e)$	$[P_{2} (T)]^{2}$	1	$2$	$2$	No
4.2.12				$\infty$	$- \infty$	$- \infty$	No

Table 4.2: Element

(P_{k} (T), P_{k} (e), [P_{k} (T)]^{2})

on triangular mesh,

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

Table 4.3 demonstrates the convergence rates for $(P_{k} (T), P_{k} (e), [P_{k + 1} (T)]^{2})$ with a stabilizer of different $j$ on triangular mesh.

element	$P_{k} (T)$	$P_{k} (e)$	$[P_{k + 1} (T)]^{2}$	$j$	$r_{1}$	$r_{2}$	Proved
4.3.1				$- 1$	$0$	$0$	No
4.3.2				0	$0$	$0$	No
4.3.3	$P_{0} (T)$	$P_{0} (e)$	$[P_{1} (T)]^{2}$	1	$0$	$0$	No
4.3.4				$\infty$	$0$	$0$	No
4.3.5				$- 1$	$1$	$2$	No
4.3.6				0	$1$	$2$	No
4.3.7	$P_{1} (T)$	$P_{1} (e)$	$[P_{2} (T)]^{2}$	1	$1$	$2$	No
4.3.8				$\infty$	$1$	$2$	Yes
4.3.9				$- 1$	$2$	$3$	No
4.3.10				0	$2$	$3$	No
4.3.11	$P_{2} (T)$	$P_{2} (e)$	$[P_{3} (T)]^{2}$	1	$2$	$3$	No
4.3.12				$\infty$	$2$	$3$	Yes

Table 4.3: Element

(P_{k} (T), P_{k} (e), [P_{k + 1} (T)]^{2})

on triangular mesh,

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

Remark 4

The WG element $(P_{k} (T), P_{k} (e), [P_{k + 1} (T)]^{2})$ performs much better on rectangular meshes than triangular meshes.

Table 4.4 demonstrates the convergence rates for $(P_{k} (T), P_{k} (e), R T_{k} (T))$ with a stabilizer of different $j$ on triangular mesh.

element	$P_{k} (T)$	$P_{k} (e)$	$R T_{k} (T)$	$j$	$r_{1}$	$r_{2}$	Proved
4.4.1				$- 1$	$0$	$0$	No
4.4.2				0	$1$	$1$	No
4.4.3	$P_{0} (T)$	$P_{0} (e)$	$R T_{0} (T)$	1	$1$	$2$	No
4.4.4				$\infty$	$1$	$2$	Yes
4.4.5				$- 1$	$1$	$2$	No
4.4.6				0	$2$	$3$	No
4.4.7	$P_{1} (T)$	$P_{1} (e)$	$R T_{1} (T)$	1	$2$	$3$	No
4.4.8				$\infty$	$2$	$3$	Yes
4.4.9				$- 1$	$2$	$3$	No
4.4.10				0	$3$	$4$	No
4.4.11	$P_{2} (T)$	$P_{2} (e)$	$R T_{2} (T)$	1	$3$	$4$	No
4.4.12				$\infty$	$3$	$4$	Yes

Table 4.4: Element

(P_{k} (T), P_{k} (e), R T_{k} (T))

on triangular mesh,

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

5 The WG elements with $ℓ < s$ on rectangular mesh

The following table demonstrates the convergence rates for $(P_{k} (T), P_{k + 1} (e), [P_{k - 1} (T)]^{2})$ with a stabilizer of different $j$ on rectangular mesh.

element	$P_{k} (T)$	$P_{k + 1} (e)$	$[P_{k - 1} (T)]^{d}$	$j$	$r_{1}$	$r_{2}$	Proved
5.1.1				$- 1$	$1$	$2$	No
5.1.2				$0$	$0.5$	$1$	No
5.1.3	$P_{1} (T)$	$P_{2} (e)$	$[P_{0} (T)]^{2}$	$1$	$0$	$0$	No
5.1.4				$\infty$	$- \infty$	$- \infty$	No
5.1.5				$- 1$	$2$	$3$	No
5.1.6				0	$1.5$	$2$	No
5.1.7	$P_{2} (T)$	$P_{3} (e)$	$[P_{1} (T)]^{2}$	$1$	$1$	$1$	No
5.1.8				$\infty$	$- \infty$	$- \infty$	No
5.1.9				$- 1$	$3$	$4$	No
5.1.10				0	$2.5$	$3$	No
5.1.11	$P_{3} (T)$	$P_{4} (e)$	$[P_{2} (T)]^{2}$	$1$	$2$	$2$	No
5.1.12				$\infty$	$- \infty$	$- \infty$	No

Table 5.1: Element

(P_{k} (T), P_{k + 1} (e), [P_{k - 1} (T)]^{2})

on rectangular mesh,

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

The following table demonstrates the convergence rates for $(P_{k} (T), P_{k + 1} (e), [P_{k} (T)]^{2})$ with a stabilizer of different $j$ on rectangular mesh.

element	$P_{k} (T)$	$P_{k + 1} (e)$	$[P_{k} (T)]^{2}$	$j$	$r_{1}$	$r_{2}$	Proved
5.2.1				$- 1$	$0$	$0$	No
5.2.2				$0$	$0.5$	$1$	No
5.2.3	$P_{0} (T)$	$P_{1} (e)$	$[P_{0} (T)]^{2}$	$1$	$0$	$0$	No
5.2.4				$\infty$	$- \infty$	$- \infty$	No
5.2.5				$- 1$	$1$	$2$	No
5.2.6				$0$	$1.5$	$2$	No
5.2.7	$P_{1} (T)$	$P_{2} (e)$	$[P_{1} (T)]^{2}$	$1$	$1$	$1$	No
5.2.8				$\infty$	$- \infty$	$- \infty$	No
5.2.9				$- 1$	$2$	$3$	No
5.2.10				$0$	$2.5$	$3$	No
5.2.11	$P_{2} (T)$	$P_{3} (e)$	$[P_{2} (T)]^{2}$	$1$	$2$	$2$	No
5.2.12				$\infty$	$- \infty$	$- \infty$	No

Table 5.2: Element

(P_{k} (T), P_{k + 1} (e), [P_{k} (T)]^{2})

on rectangular mesh,

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

Table 5.3 demonstrates the convergence rates for $(P_{k} (T), P_{k + 1} (e), [P_{k + 1} (T)]^{2})$ with a stabilizer of different $j$ on rectangular mesh.

element	$P_{k} (T)$	$P_{k + 1} (e)$	$[P_{k + 1} (T)]^{2}$	$j$	$r_{1}$	$r_{2}$	Proved
5.3.1				$- 1$	$0$	$0$	No
5.3.2				$0$	$0.5$	$1$	No
5.3.3	$P_{0} (T)$	$P_{1} (e)$	$[P_{1} (T)]^{2}$	$1$	$1$	$2$	No
5.3.4				$\infty$	$1$	$2$	No
5.3.5				$- 1$	$1$	$2$	No
5.3.6				$0$	$1.5$	$3$	No
5.3.7	$P_{1} (T)$	$P_{2} (e)$	$[P_{2} (T)]^{2}$	$1$	$2$	$4$	No
5.3.8				$\infty$	$2$	$4$	No
5.3.9				$- 1$	$2$	$3$	No
5.3.10				$0$	$2.5$	$4$	No
5.3.11	$P_{2} (T)$	$P_{3} (e)$	$[P_{3} (T)]^{2}$	$1$	$3$	$5$	No
5.3.12				$\infty$	$3$	$5$	No

Table 5.3: Element

(P_{k} (T), P_{k + 1} (e), [P_{k + 1} (T)]^{2})

on rectangular mesh,

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

Table 5.4 demonstrates the convergence rates for $(P_{k} (T), P_{k + 1} (e), [P_{k + 2} (T)]^{2})$ with a stabilizer of different $j$ on rectangular mesh.

element	$P_{k} (T)$	$P_{k + 1} (e)$	$[P_{k + 2} (T)]^{2}$	$j$	$r_{1}$	$r_{2}$	Proved
5.4.1				$- 1$	$0$	$0$	No
5.4.2				$0$	$0$	$0$	No
5.4.3	$P_{0} (T)$	$P_{1} (e)$	$[P_{2} (T)]^{2}$	$1$	$0$	$0$	No
5.4.4				$\infty$	$0$	$0$	No
5.4.5				$- 1$	$1$	$2$	No
5.4.6				$0$	$1$	$2$	No
5.4.7	$P_{1} (T)$	$P_{2} (e)$	$[P_{3} (T)]^{2}$	$1$	$1$	$2$	No
5.4.8				$\infty$	$1$	$2$	No
5.4.9				$- 1$	$2$	$3$	No
5.4.10				$0$	$2$	$3$	No
5.4.11	$P_{2} (T)$	$P_{3} (e)$	$[P_{4} (T)]$

Numerical investigation on weak Galerkin finite elements

Authors

De Rham Complexes for Weak Galerkin Finite Element Spaces

A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction

The lowest-order stabilizer free Weak Galerkin Finite Element Method

Generalized Weak Galerkin Methods For Stokes Equations

Quadrature-free Immersed Isogeometric Analysis

The Source Stabilized Galerkin Formulation for Linear Moving Conductor Problems with Edge Elements

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

1 Introduction

2 Weak Galerkin Finite Element formulations

Definition 1

Definition 2

Remark 1

Definition 3

Weak Galerkin Algorithm 1

3 The WG elements with $ℓ = s$ on rectangular mesh

Remark 2

Remark 3

4 The WG elements for $ℓ = s$ on triangular mesh

Remark 4

5 The WG elements with $ℓ < s$ on rectangular mesh

Numerical investigation on weak Galerkin finite elements

Authors

Related Research

De Rham Complexes for Weak Galerkin Finite Element Spaces

A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction

The lowest-order stabilizer free Weak Galerkin Finite Element Method

Generalized Weak Galerkin Methods For Stokes Equations

Quadrature-free Immersed Isogeometric Analysis

The Source Stabilized Galerkin Formulation for Linear Moving Conductor Problems with Edge Elements

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

This week in AI

1 Introduction

2 Weak Galerkin Finite Element formulations

Definition 1

Definition 2

Remark 1

Definition 3

Weak Galerkin Algorithm 1

3 The WG elements with ℓ=s on rectangular mesh

Remark 2

Remark 3

4 The WG elements for ℓ=s on triangular mesh

Remark 4

5 The WG elements with ℓ<s on rectangular mesh

3 The WG elements with $ℓ = s$ on rectangular mesh

4 The WG elements for $ℓ = s$ on triangular mesh

5 The WG elements with $ℓ < s$ on rectangular mesh