The weak Galerkin method for eigenvalue problems

The weak Galerkin method for eigenvalue problems

Hehu Xie LSEC and Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China (hhxie@lsec.cc.ac.cn)    Qilong Zhai Department of Mathematics, Jilin University, Changchun, China (diql15@mails.jlu.edu.cn).    Ran Zhang Department of Mathematics, Jilin University, Changchun, China (zhangran@mail.jlu.edu.cn). The research of Zhang was supported in part by China Natural National Science Foundation(11271157, 11371171, 11471141), and by the Program for New Century Excellent Talents in University of Ministry of Education of China.
Abstract

This article is devoted to computing the eigenvalue of the Laplace eigenvalue problem by the weak Galerkin (WG) finite element method with emphasis on obtaining lower bounds. The WG method 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 leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. We establish the optimal-order error estimates for the WG finite element approximation for the eigenvalue problem. Comparing with the classical nonconforming finite element method which can just provide lower bound approximation by linear elements with only the second order convergence, the WG methods can naturally provide lower bound approximation with a high order convergence (larger than ). Some numerical results are also presented to demonstrate the efficiency of our theoretical results.

Key words. weak Galerkin finite element methods, eigenvalue problem, lower bound, error estimate, finite element method.

AMS subject classifications. Primary, 65N30, 65N15, 65N12, 74N20; Secondary, 35B45, 35J50, 35J35

1 Introduction

The study of eigenvalues and eigenfunctions of partial differential operators both in theoretical and approximation grounds is very important in many fields of sciences, such as quantum mechanics, fluid mechanics, stochastic process, structural mechanics, etc. Thus, a fundamental work is to find the eigenvalues and corresponding eigenfunctions of partial differential operators.

In this paper, we consider the following model problem: Find such that

\hb@xt@.01(1.1)

where is a polyhedral domain in . For simplicity, we are only concerned with the case , while all the conclusions can be extended to trivially. The classical variational form of problem (LABEL:problem-eq) is defined as follows: Find and such that and

\hb@xt@.01(1.2)

where

It is well known that the problem (LABEL:vartion-form) has the eigenvalue sequence [2]

with the corresponding eigenfunction sequence

which satisfies the property ( denotes the Kronecker function).

The first aim of this paper is to analyze the weak Galerkin finite element method for the eigenvalue problem. We will give the corresponding error estimates for the eigenpair approximations by the weak Galerkin finite element method based on the standard theory from [2, 3].

The eigenvalue is a positive real number, and thus it is credible if we get both the upper and lower bounds. In fact, a simple combination of lower and upper bounds will present intervals to which exact eigenvalue belongs. For the Laplace eigenvalue problems, since the Rayleigh quotient and minimum-maximum principle, it is easily to obtain the upper bounds of eigenvalue by any standard conforming finite element methods [6, 26]. For the lower bounds, the computation is of high interest and generally more difficult. Influenced by the minimum-maximum principle, people try to obtain the lower bounds with the nonconforming element methods. In fact, the lower bound property of eigenvalues by nonconforming elements are observed in numerical aspects at the beginning [3, 13, 24, 25, 30, 33]. After that, a series of results make progress in this aspect, e.g., Lin and Lin [12] proved that the non-conforming rectangular element approximates exact eigenvalues associated with smooth eigenfunctions from below in 2006. Hu, Huang and Shen [8] gets the lower bound of Laplace eigenvalue problem by conforming linear and bilinear elements together with the mass lumping method. A general kind of expansion method, which was first proved in its full term in [31] by the similar argument in [1], is extensively used in [7, 10, 16] and the references cited therein. Another interesting way is provided in [14] and the corresponding paper cited therein.

Our work was inspired by some recent studies of lower end approximation of eigenvalues by finite element discretization for some elliptic partial differential operators [1, 7, 16, 11, 31]. One crucial technical ingredient that is needed in the analysis is some lower bound of the eigenfunction discretization error by the finite element method. The mainly challenge aspect is to design the high order elements to present the lower bound scheme. So far, the lower bound finite element methods are mainly first order nonconforming finite element methods. It is desired to design some high order numerical methods to obtain the lower bound eigenvalue approximations which is the second aim of this paper.

Recently a new class of finite element methods, called weak Galerkin (WG) finite element methods have been developed for the partial differential equations for its highly flexible and robust properties. The WG method refers to a numerical scheme for partial differential equations in which differential operators are approximated by weak forms as distributions over a set of generalized functions. It has been demonstrated that the WG method is highly flexible and robust as a numerical technique that employs discontinuous piecewise polynomials on polygonal or polyhedral finite element partitions. This thought was first proposed in [28] for a model second order elliptic problem in 2012, and further developed in [9, 18, 19, 20, 21, 22, 23, 29, 32] with other applications. In order to enforce necessary weak continuities for approximating functions, proper stabilizations are employed. The main advantages of the WG method include the finite element partition can be of polytopal type with certain regular requirements, the weak finite element space is easy to construct with any given approximation requirement and the WG schemes can be hybridized so that some unknowns associated with the interior of each element can be locally eliminated, yielding a system of linear equations involving much less number of unknowns than what it appears.

The objective of the present paper is twofold. First, we will introduce weak Galerkin method for solving the Laplacian eigenvalue problem, which has the optimal-order error estimates. Furthermore we investigate the performance of the WG methods for presenting the guaranteed lower bound approximation of the eigenvalues problem under the assumption that the global mesh-size is sufficiently small. To demonstrate the potential of WG finite element methods in solving eigenvalue problems, we will restrict ourselves to any order WG elements (even for higher order WG finite element spaces) and investigate the robustness and effectiveness of this method.

An outline of the paper is as follows. In Section 2, the necessary notations, definitions of weak functions and weak derivatives are introduced. The WG finite element scheme of the Laplace eigenvalue problem is stated. Error estimates for the boundary value problem are presented in Section 3. In Section 4, we establish the error estimates for the WG finite element approximation for the eigenvalue problem. Section 5 is devoted to presenting any higher order accuracy lower bound approximation of the eigenvalues. Some numerical results are presented in Section 6 to demonstrate the efficiency of our theoretical results and some concluding remarks are given in the last section.

2 The weak Galerkin scheme for the eigenvalue problem

2.1 Preliminaries and notations

First, we present some notation which will be used in this paper. We denote and the inner-product and the norm on . If the region is an edge or boundary of some element, we use instead of . We shall drop the subscript when or . In this paper, denotes the space of polynomials on with degree no more than . Throughout this paper, denotes a generic positive constant which is independent of the mesh size.

Let be a partition of the domain , and the elements in are polygons satisfying the regular assumptions specified in [29]. Denote by the edges in , and by the interior edges . For each element , represents the diameter of , and denotes the mesh size.

2.2 A weak Galerkin scheme

Now we introduce a weak Galerkin scheme solving the problem (LABEL:problem-eq). For a given integer , define the Weak Galerkin (WG) finite element space

For each weak function , we can define its weak gradient by distribution element-wisely as follows.

Definition 2.1

For each , is the unique polynomial in satisfying

\hb@xt@.01(2.1)

where denotes the outward unit normal vector.

For the aim of analysis, some projection operators are also employed in this paper. Let denote the projection from onto , denote the projection from onto , and denote the projection from onto . Combining and together, we can define , which is a projection from onto .

Now we define three bilinear forms on that for any ,

where is a small constant parameter to be selected. With these preparations we can give the following weak Galerkin algorithm.

Weak Galerkin Algorithm 1

Find , such that and

\hb@xt@.01(2.2)

3 Error estimates for the boundary value problem

In order to analyze the error of the eigenvalue problem by the weak Galerkin method, we need some estimates for the boundary value problem. The main idea is similar to [27] but some modifications.

3.1 A weak Galerkin method for the Poisson equation

In this section, we consider the weak Galerkin method for the following Poisson equation

\hb@xt@.01(3.1)

The corresponding weak Galerkin scheme is to find such that

\hb@xt@.01(3.2)

Define a semi-norm on as follows

We claim that is indeed a norm on . In order to check the positive property, suppose . Then we have in and on for all . It follows that

so that is piecewise constant and on . Notice that on , we can obtain that . For the analysis, we also define another norm on as

Furthermore, it is easy to check that the weak Galerkin scheme (LABEL:WG-scheme1) is symmetric and positive definite, which has a unique solution.

The following commutative property plays an essential role in the forthcoming proof, which shows that the weak gradient operator is an approximation of the classical gradient operator.

Lemma 3.1

([27]) For any element , the following commutative property holds true,

\hb@xt@.01(3.3)

Proof. From the definition of the weak gradient (LABEL:def-wgradient) and the integration by parts, we have that for any

which completes the proof.     

3.2 Error equation

Suppose is the solution of (LABEL:Poisson-eq1), and is the numerical solution of (LABEL:WG-scheme1). Denote by the error that

Then should satisfy the following equation.

Lemma 3.2

Let be the error of the weak Galerkin scheme (LABEL:WG-scheme1). Then, for any , we have

\hb@xt@.01(3.4)

where

Proof. From the definition of the weak gradient (LABEL:def-wgradient) and the commutative property (LABEL:commu-prop), we can obtain on each element that

Summing over all elements and it follows that

Notice that the numerical solution satisfies (LABEL:WG-scheme1). Then we can derive that

which completes the proof.     

In order to estimate the right hand side terms of (LABEL:error-eqn), we still need some technique tools introduced in [29].

Lemma 3.3

([29])  (Trace Inequality) Let be a partition of the domain into polygons in 2D or polyhedra in 3D. Assume that the partition satisfies the Assumptions A1, A2, and A3 as stated in [29]. Let be any real number. Then, there exists a constant such that for any and edge/face , we have

\hb@xt@.01(3.5)

for any .

Lemma 3.4

([29]) (Inverse Inequality) Let be a partition of the domain into polygons or polyhedra. Assume that satisfies all Assumptions A1-A4 and be any real number. Then, there exists a constant such that

\hb@xt@.01(3.6)

for any piecewise polynomial of degree no more than on .

Lemma 3.5

([29]) Let be a finite element partition of satisfying the shape regularity assumptions specified in [29] and . Then, for we have

\hb@xt@.01(3.7)
\hb@xt@.01(3.8)

where denotes a generic constant independent of mesh size and the functions in the estimates.

Suppose and let . With the tools above we can give the estimates for and as follows.

Lemma 3.6

For each element , we have

Furthermore, there is

Proof. From the trace inequality (LABEL:Trace_inequality00) and the definition of the weak gradient operator (LABEL:def-wgradient), we have following inequalities for any ,

which implies that

Applying the Poincaré inequality, we can obtain

which completes the proof.     

Lemma 3.7

For any and , the following estimates hold true,

where .

Proof. From the Cauchy-Schwarz inequality and Lemma LABEL:norm-equi1, we can obtain

Similarly, for the second term we can derive that

which completes the proof.     

3.3 Error estimates

With the error equation (LABEL:error-eqn) and the estimates derived in Lemma LABEL:remainder, we can get the following error estimate for the weak Galerkin method.

Theorem 3.8

Assume the exact solution of (LABEL:Poisson-eq1), , and is the numerical solution of the weak Galerkin scheme (LABEL:WG-scheme1). Denote , then the following estimates hold true,

\hb@xt@.01(3.9)
\hb@xt@.01(3.10)

Proof. Taking in (LABEL:error-eqn) and it follows that

From the definition of , we can easily get that when is small,

which completes the proof.     

Using a standard dual argument, which is similar to the technique applied in [27], and then we can obtain the following error estimate.

Theorem 3.9

Assume the exact solution of (LABEL:Poisson-eq1), , and is the numerical solution of the weak Galerkin scheme (LABEL:WG-scheme1). In addition, assume the dual problem has -regularity. Denote , then the following estimate holds true

\hb@xt@.01(3.11)

4 Error estimates for the eigenvalue problem

In this section, we turn back to the approximation of the eigenvalue problem (LABEL:problem-eq). Denote , and define the sum space . Now we introduce the following semi-norm on that

We claim that indeed defines a norm on . For any , if is defined in the sense of trace, we shall show that is equivalent to , which defines a norm on .

Lemma 4.1

is equivalent to on .

Proof. It is obvious that for any ,

Then we only need to show that

To this end, denote by the projection onto , and it follows the trace inequality (LABEL:Trace_inequality00) and the Poincare’s inequality that

which completes the proof.     

As to the space , we have the following equivalence lemma.

Lemma 4.2

([27]) There exists two constants and such that for any , we have

\hb@xt@.01(4.1)

i.e. and are equivalent on .

Proof. In order to prove the equivalence, we just need to verify the following inequalities that for any ,

\hb@xt@.01(4.2)
\hb@xt@.01(4.3)

The inequality (LABEL:norm-neq1) has been proved in Lemma LABEL:norm-equi1. For handling the inequality (LABEL:norm-neq2), we use the definition of the weak gradient to get that

Then we can derive from Lemma LABEL:norm-equi1 that

which completes the proof.     

Now we define two operators and as follows

The following lemmas show that the finite element space and the discrete solution operator are approximations of and .

Lemma 4.3

Suppose , then we have

Proof. From the trace inequality (LABEL:Trace_inequality00) and Lemma LABEL:projection-prop we have

which completes the proof.     

As we know, we can extend the operators and to the operators from to which will not change the non-zero spectrums of the operators and .

Lemma 4.4

The operators and have the following estimate

where denote the operator norm from to .

Proof. Since is a Hilbert space, it is equivalent to verify that

For any with , suppose and . From the error estimate (LABEL:err-est2) and the regularity of the Poisson’s equation, we have

Then the equivalence Lemma LABEL:norm-equi implies that

Moreover, by letting in Lemma LABEL:interpolation we can obtain that

It follows the triangle inequality that

Notice that , which completes the proof.     

Lemma 4.5

The operator is compact.

Proof. Denote the restriction of on . Since is finite dimensional, is compact. Notice that , so . In order to prove that is compact, we just need to verify that is bounded.

For any , . For , we can conclude from Lemma LABEL:interpolation that

which completes the proof.     

Now we review some notations in the spectral approximation theory. We denote by the spectrum of , and by the resolvent set. represents the resolvent operator. Let be a nonzero eigenvalue of with algebraic multiplicities . Let be a circle in the complex plane centered at which lies in and encloses no other points of . The corresponding spectral projection is

represents the range of , which is the space of generalized eigenvectors.

For a Banach space and its closed subspaces and , define the distances as follows that

Lemma 4.6

Suppose the eigenvectors of   have -regularity, i.e. for any . Denote , then the following estimate holds true,

Proof. Suppose with . Similar to the proof of Lemma LABEL:operator-approx, from Theorem LABEL:err-est we have

From Lemma LABEL:interpolation we can obtain

which implies

Since is finite dimensional, there is a uniform upper bound for , where with , which completes the proof.     

For the symmetry of and , and are self-adjoint. In addition, if we change the norm to norm, all the conclusions in this section can be interpreted trivially. Then from the theory in [2, 4], we can derive the following estimates.

Theorem 4.7

Suppose is the -th eigenvalue of (LABEL:WG-scheme) and is the corresponding eigenvector. There exist an exact eigenvalue