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

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous approximations, which maintains simple finite element formulation. Stabilizer free weak Galerkin methods further simplify the WG methods and reduce computational complexity. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the stabilizer free WG schemes without compromising the accuracy of the numerical approximation. A new stabilizer free weak Galerkin finite element method is proposed and analyzed with polynomial degree reduction. To achieve such a goal, a new definition of weak gradient is introduced. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete H^1 norm and the standard L^2 norm. The numerical examples are tested on various meshes and confirm the theory.

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1 Introduction

The weak Galerkin finite element method is an effective and flexible numerical technique for solving partial differential equations. The main idea of weak Galerkin finite element methods is the use of weak functions and their corresponding discrete weak derivatives in algorithm design. The WG method was first introduced in [13, 14] and then has been applied to solve various partial differential equations [2, 4, 3, 5, 7, 8, 9, 10, 6, 11, 12, 15].

A stabilizing/penalty term is often essential in finite element methods with discontinuous approximations to enforce connection of discontinuous functions across element boundaries. Removing stabilizers from discontinuous finite element methods simplifies finite element formulations and reduces programming complexity. Stabilizer free WG finite element methods have been studied in [16, 1, 20]. The idea is increasing the connectivity of a weak function across element boundary by raising the degree of polynomials for computing weak derivatives. In [16], we have proved that a stabilizer can be removed from the WG finite element formulation for the WG element if , where is the number of edges/faces of an element. The condition has been relaxed in [1]. Stabilizer free DG methods have also been developed in [17, 18].

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 , denotes the gradient of the function , and is a symmetric matrix-valued function in . We shall assume that there exist two positive numbers such that

(1.3)

Here

is understood as a column vector and

is the transpose of .

The goal of this paper is to propose and analyze a stabilizer free weak Galerkin method for (1.1)-(1.2) by using less number of unknowns than that of [16] without compromising the order of convergence. The WG scheme will use the configuration of , . For the WG element , , the stabilizer free WG method with the standard definition of weak gradient only produces suboptimal convergence rates in both energy norm and the norm, shown in Table 1.1 ([19]).

Table 1.1: Weak gradient calculated by (1.4), and .

The standard definition for a weak gradient of a weak function is a piecewise vector valued polynomial such that on each , satisfies ([13, 14, 8])

(1.4)

A new way of defining weak gradient is introduced in (2.3) for the WG element such that the our new corresponding stabilizer free WG approximation converges to the true solution with optimal order convergence rates, shown in Table 1.2.

Table 1.2: Weak gradient calculated by (2.3), and .

We also prove the optimal convergence rates theoretically for the stabilizer free WG approximation in an energy norm and in the norm. The numerical examples are tested on different finite element partitions.

2 Weak Galerkin Finite Element Schemes

Let be a partition of the domain consisting of polygons in two dimension or polyhedra in three dimension satisfying a set of conditions specified in [14]. Denote by the set of all edges/faces in , and let be the set of all interior edges/faces. For simplicity, we will use term edge for edge/face without confusion.

For a given integer , let be the weak Galerkin finite element space associated with defined as follows

(2.1)

and its subspace is defined as

(2.2)

We would like to emphasize that any function has a single value on each edge .

For , a weak gradient is a piecewise vector valued polynomial such that on each , satisfies

(2.3)

where depends on the shape of the elements and will be determined later. In the above equation, we let and if .

For simplicity, we adopt the following notations,

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 .

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)

The following lemma reveals a nice property of the weak gradient. Let , then on any , we have

(2.5)

The definition of weak gradient (2.3) implies that for any

which implies (2.5).

For any function , the following trace inequality holds true (see [14] for details):

(2.6)

3 Well Posedness

For any , define two semi-norms

(3.1)
(3.2)

It follows from (1.3) that there exist two positive constants and such that

(3.3)

We introduce a discrete semi-norm as follows:

(3.4)

It is easy to see that defines a norm in .

Next we will show that also defines a norm for by proving the equivalence of and in . First we need the following lemma.

([18]) Let be a convex, shape-regular -polygon/polyhedron of size . Let be an edge/face-polygon of , of size . Let and on . Then there exists a polynomial , , such that

(3.5)
(3.6)
(3.7)

There exist two positive constants and such that for any , we have

(3.8)

For any , it follows from the definition of weak gradient (2.3) and integration by parts that on each

(3.9)

By letting in (3.9) we arrive at

Letting in (3.9) implies

(3.10)

From the trace inequality (2.6) and the inverse inequality we have

which implies

and consequently

Next we will prove . First we need to prove

(3.11)

For and with , it has been proved in Lemma 3 that there exists such that

(3.12)

and

(3.13)

Substituting into (3.9), we get

(3.14)

It follows from Cauchy-Schwarz inequality and (3.13) that

which implies

(3.15)

It follows from (3.10), the trace inequality, the inverse inequality and (3.15),

which implies

Combining the above estimate and (3.15), we prove the lower bound of (3.8) and complete the proof of the lemma.

The weak Galerkin finite element scheme (2.4) has a unique solution.

If and are two solutions of (2.4), then would satisfy the following equation

Then by letting in the above equation and (3.3), we arrive at

It follows from (3.8) that . Since is a norm in , one has . This completes the proof of the lemma.

4 Error Analysis

The goal of this section is to establish error estimates for the weak Galerkin finite element solution arising from (2.4

). For simplicity of analysis, we assume that the coefficient tensor

in (1.1) is a piecewise constant matrix with respect to the finite element partition . The result can be extended to variable tensors without any difficulty, provided that the tensor is piecewise sufficiently smooth.

4.1 Error Equation

Let and . In this section, we derive an error equation that satisfies.

For any , the following error equation holds true

(4.1)

where

For , testing (1.1) by and using the fact that , we arrive at

(4.2)

Obviously, we have

(4.3)

Combining (4.2) and (4.3) gives

(4.4)

It follows from (2.3) and (2.5) that

(4.5)

Using (4.4) and (4.5), we have

(4.6)

Subtracting (2.4) from (4.6) yields,

This completes the proof of the lemma.

4.2 Error Estimates in Energy Norm

Optimal convergence rate of the WG approximation in energy norm will be obtained in this section. First we will bound the two terms and in the error equation (4.1).

For any and , we have

(4.7)
(4.8)

Using the Cauchy-Schwarz inequality, the trace inequality (2.6), (1.3) and (3.8), we have

Let be the element-wise defined projection onto on each . Using the Cauchy-Schwarz inequality, the trace inequality (2.6), (1.3) and the inverse inequality, we have

We have proved the lemma.

Let , then

(4.9)

It follows from (2.3) and (2.6),

Letting in the above equation and taking summation over , we have

We have proved the lemma.

Let be the weak Galerkin finite element solution of (2.4). Assume the exact solution . Then, there exists a constant such that

(4.10)

It is straightforward to obtain

We will bound the two terms in (4.2). Letting in (4.1) and using (4.7), (4.8) and (4.9), we have

(4.12)

The estimate (4.9) implies

(4.13)

Combining the estimates (4.12) and (4.13) with (4.2), we arrive

which completes the proof.

The estimates (4.9) and (4.10) imply

(4.14)

4.3 Error Estimates in Norm

The standard duality argument is used to obtain error estimate. Recall and . The dual problem seeks satisfying

(4.15)

Assume that the following -regularity holds

(4.16)

Let be the weak Galerkin finite element solution of (2.4). Assume that the exact solution and (4.16) holds true. Then, there exists a constant such that

(4.17)

By testing (4.15) with we obtain

(4.18)

where we have used the fact that on . Setting and in (4.5) yields

(4.19)

Substituting (4.19) into (4.18) gives

(4.20)