The Weak Galerkin Method for Elliptic Eigenvalue Problems

Abstract

This article is devoted to studying the application of the weak Galerkin (WG) finite element method to the elliptic eigenvalue problem with an emphasis on obtaining lower bounds. The WG method uses discontinuous polynomials on polygonal or polyhedral finite element partitions. The non-conforming finite element space of the WG method is the key of the lower bound property. It also makes the WG method more robust and flexible in solving eigenvalue problems. We demonstrate that the WG method can achieve arbitrary high convergence order. This is in contrast with existing nonconforming finite element methods which can provide lower bound approximations by linear finite elements. Numerical results are presented to demonstrate the efficiency and accuracy of the theoretical results.

An Algebraic Multigrid Method for Eigenvalue Problems and Its Numerical Tests

“…Therefore, the construction of efficient eigensolvers with a nearly optimal computational complexity is very important. It is natural to use AMG and MG methods in eigenvalue problems [3,8,13,14,28,35,37]. A good survey of various application of the AMG methods in eigenvalue problems is presented in [17].…”

Computation of Eigenvalues for Nonlocal Models by Spectral Methods

“…Several numerical methods are proposed to study the Laplacian eigenvalue problem. In particular the weak Galerkin finite element method and its accelerated version such as the two-grid and two-space methods are highly flexible and efficient methods used for the computation of the Laplacian eigenvalues, (see [30,29]).…”

A locking-free discontinuous Galerkin method for linear elastic Steklov eigenvalue problem

“…In [36], the authors proposed the weak Galerkin finite element method for elliptic eigenvalue problems. A general framework was proposed and applied to elliptic eigenvalue problems.…”

“…A general framework was proposed and applied to elliptic eigenvalue problems. As long as (A1)-(A7) in [36] are proved, the eigenvalues obtained by the WG method are asymptotic lower bounds of the exact eigenvalues. Recently, Carstensen, Zhai and Zhang [8] proposed a skeletal finite element method that can compute lower eigenvalue bounds.…”