Numerical investigation on weak Galerkin finite elements

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. The novel idea of weak Galerkin finite element methods is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces generates different weak Galerkin finite elements. The purpose of this paper is to study stability, convergence and supercloseness of different WG elements by providing many numerical experiments recorded in 31 tables. These tables serve two purposes. First it provides a detail guide of the performance of different WG elements. Second, the information in the tables opens new research territory why some WG elements outperform others.

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1 Introduction

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

(1.1)
(1.2)

where is a polytopal domain in .

The weak form of the problem (1.1)-(1.2) is to find such that on and satisfies

(1.3)

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 with inside of each element and on the boundary of the element. Both and can be approximated by polynomials in and respectively, where stands for an element and the edge or face of , and are non-negative integers with possibly different values. Weak derivatives are defined for weak functions in the sense of distributions. Denote by

the vector space for weak gradient. Typical choices for

are or . Various combination of 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 such that on and satisfies

(1.4)

where is a parameter independent stabilizer. Another one is WG stabilizer free formulation [1, 20, 21]: find such that on and satisfies

(1.5)

Removing stabilizers simplifies the formulations and reduces programming complexity. A stabilizer free WG method can be obtained by raising the degree of polynomial for approximating weak gradient in the WG element .

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 has two orders of supercloseness in both energy norm and norm on rectangular meshes, shown in Table 3.3. It is proved in [2] that the WG element has two orders of supercloseness in both energy norm and norm, on general triangular meshes in Table 6.3. Due to the bad behavior of the WG element 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 be a partition of the domain consisting of rectangles or triangles. Denote by the set of all edges in , and let be the set of all interior edges or flat faces. For every element , we denote by its diameter and mesh size for . Let consist all the polynomials defined on of degree less or equal to .

Definition 1

For and , define a local WG element as,

(2.1)
Definition 2

For any , a weak gradient is defined as a unique solution of the following equation

(2.2)

A typical choice of is , or . Different combinations of associated with a WG element leads to different weak Galerkin finite element formulations. The weak gradient defined in (2.2) is an approximation of that is computed on each element .

Remark 1

Please note that the space

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 associated with as follows

(2.3)

We would like to emphasize that any function has a single value on each edge . The subspace of consisting of functions with vanishing boundary value is denoted as .

Let and be the two element-wise defined projections onto and on each , respectively. Define . Let be the element-wise defined projection onto on each element .

For simplicity, we adopt the following notations,

Weak Galerkin Algorithm 1

A numerical approximation for (1.1)-(1.2) can be obtained by seeking satisfying on and the following equation:

(2.4)

where the stabilizer is defined as

(2.5)

Let in (2.5), we mean , i.e., we have the following stabilizer-free WG formulation,

(2.6)

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, refers to the stabilizer free WG formulation (2.6), where is defined in (2.5).

3 The WG elements with on rectangular mesh

Next we will study convergence rate for the WG element with 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 with a stabilizer of different defined in (2.5) on rectangular mesh.

element Proved
3.1.1 Yes
3.1.2 No
3.1.3 No
3.1.4 No
3.1.5 Yes
3.1.6 No
3.1.7 No
3.1.8 No
3.1.9 Yes
3.1.10 No
3.1.11 No
3.1.12 No
Table 3.1: Element on rectangular mesh, and .

Table 3.2 demonstrates the convergence rates for with a stabilizer of different on rectangular mesh.

element Proved
3.2.1 No
3.2.2 No
3.2.3 No
3.2.4 No
3.2.5 No
3.2.6 No
3.2.7 No
3.2.8 No
3.2.9 No
3.2.10 No
3.2.11 No
3.2.12 No
Table 3.2: Element on rectangular mesh, and .
Remark 2

Theorem 4.9 in [16] guarantees the optimal convergence rate of the WG element 3.2.5 in the norm. However the optimal convergence rate in the norm is not proved in [16]. Therefore, we still mark in the case 3.2.5.

Table 3.3 demonstrates the convergence rates for with a stabilizer of different on rectangular mesh.

element Proved
3.3.1 No
3.3.2 No
3.3.3 No
3.3.4 No
3.3.5 No
3.3.6 No
3.3.7 No
3.3.8 Yes
3.3.9 No
3.3.10 No
3.3.11 No
3.3.12 Yes
Table 3.3: Element on rectangular mesh, and .

Table 3.4 demonstrates the convergence rates for with a stabilizer of different on rectangular mesh.

element Proved
3.4.1 No
3.4.2 No
3.4.3 No
3.4.4 No
3.4.5 No
3.4.6 No
3.4.7 No
3.4.8 Yes
3.4.9 No
3.4.10 No
3.4.11 No
3.4.12 Yes
Table 3.4: Element on rectangular mesh, and .
Remark 3

For the 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 to . Specially the WG element shows order two supercloseness in Table 3.3. However, for the element , the numerical tests in Table 3.4 show the convergence rate of the WG solution decreasing. Remember that increasing in for weak gradient does not introduce additional degrees of freedom for the resulting linear systems.

Table 3.5 demonstrates the convergence rates for with a stabilizer of different on rectangular mesh.

element Proved
3.5.1 No
3.5.2 0 No
3.5.3 1 No
3.5.4 No
3.5.5 No
3.5.6 0 No
3.5.7 1 No
3.5.8 No
3.5.9 No
3.5.10 0 No
3.5.11 1 No
3.5.12 No
Table 3.5: Element on rectangular mesh, and .

4 The WG elements for 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 with a stabilizer of different on triangular mesh.

element Proved
4.1.1 Yes
4.1.2 0 No
4.1.3 1 No
4.1.4 No
4.1.5 Yes
4.1.6 0 No
4.1.7 1 No
4.1.8 No
4.1.9 Yes
4.1.10 0 No
4.1.11 1 No
4.1.12 No
Table 4.1: Element on triangular mesh, and .

Table 4.2 demonstrates the convergence rates for with a stabilizer of different on triangular mesh.

element Proved
4.2.1 No
4.2.2 0 No
4.2.3 1 No
4.2.4 No
4.2.5 No
4.2.6 0 No
4.2.7 1 No
4.2.8 No
4.2.9 No
4.2.10 0 No
4.2.11 1 No
4.2.12 No
Table 4.2: Element on triangular mesh, and .

Table 4.3 demonstrates the convergence rates for with a stabilizer of different on triangular mesh.

element Proved
4.3.1 No
4.3.2 0 No
4.3.3 1 No
4.3.4 No
4.3.5 No
4.3.6 0 No
4.3.7 1 No
4.3.8 Yes
4.3.9 No
4.3.10 0 No
4.3.11 1 No
4.3.12 Yes
Table 4.3: Element on triangular mesh, and .
Remark 4

The WG element performs much better on rectangular meshes than triangular meshes.

Table 4.4 demonstrates the convergence rates for with a stabilizer of different on triangular mesh.

element Proved
4.4.1 No
4.4.2 0 No
4.4.3 1 No
4.4.4 Yes
4.4.5 No
4.4.6 0 No
4.4.7 1 No
4.4.8 Yes
4.4.9 No
4.4.10 0 No
4.4.11 1 No
4.4.12 Yes
Table 4.4: Element on triangular mesh, and .

5 The WG elements with on rectangular mesh

The following table demonstrates the convergence rates for with a stabilizer of different on rectangular mesh.

element Proved
5.1.1 No
5.1.2 No
5.1.3 No
5.1.4 No
5.1.5 No
5.1.6 0 No
5.1.7 No
5.1.8 No
5.1.9 No
5.1.10 0 No
5.1.11 No
5.1.12 No
Table 5.1: Element on rectangular mesh, and .

The following table demonstrates the convergence rates for with a stabilizer of different on rectangular mesh.

element Proved
5.2.1 No
5.2.2 No
5.2.3 No
5.2.4 No
5.2.5 No
5.2.6 No
5.2.7 No
5.2.8 No
5.2.9 No
5.2.10 No
5.2.11 No
5.2.12 No
Table 5.2: Element on rectangular mesh, and .

Table 5.3 demonstrates the convergence rates for with a stabilizer of different on rectangular mesh.

element Proved
5.3.1 No
5.3.2 No
5.3.3 No
5.3.4 No
5.3.5 No
5.3.6 No
5.3.7 No
5.3.8 No
5.3.9 No
5.3.10 No
5.3.11 No
5.3.12 No
Table 5.3: Element on rectangular mesh, and .

Table 5.4 demonstrates the convergence rates for with a stabilizer of different on rectangular mesh.

element Proved
5.4.1 No
5.4.2 No
5.4.3 No
5.4.4 No
5.4.5 No
5.4.6 No
5.4.7 No
5.4.8 No
5.4.9 No
5.4.10 No
5.4.11