A Class of Spectral Element Methods and Its A Priori/A Posteriori Error Estimates for 2nd-Order Elliptic Eigenvalue Problems

Jiayu Han,Yidu Yang

doi:10.1155/2013/262010

Abstract

This paper discusses spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis for general 2nd-order elliptic eigenvalue problems. The special work of this paper is as follows. (1) We prove a priori and a posteriori error estimates for spectral and spectral element methods. (2) We compare between spectral methods, spectral element methods, finite element methods and their derivedp-version,h-version, andhp-version methods from accuracy, degree of freedom, and stability and verify that spectral methods and spectral element methods are highly efficient computational methods.

Highlights

As we know, finite element methods are local numerical methods for partial differential equations and well suitable for problems in complex geometries, whereas spectral methods can provide a superior accuracy, at the expense of domain flexibility
Spectral element methods combine the advantages of the above methods
Spectral and spectral element methods are widely applied to boundary value problems, as well as applied to symmetric eigenvalue problems

Summary

Introduction

Finite element methods are local numerical methods for partial differential equations and well suitable for problems in complex geometries, whereas spectral methods can provide a superior accuracy, at the expense of domain flexibility. A posteriorii error estimates and highly efficient computational methods for finite elements of eigenvalue problems are the subjects focused on by the academia these years; see [3,4,5,6,7,8,9,10,11,12,13,14,15,16], and among them, for nonsymmetric 2nd-order elliptic eigenvalue problems, [5, 15] provide a posteriori error estimates and adaptive algorithms, [9] the function value recovery techniques and [8, 10] two-level discretization schemes. (1) We prove a priori and a posteriori error estimates of spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis, respectively, for the general 2nd-order elliptic eigenvalue problems.

Preliminaries

A Posteriori Error Estimates

Numerical Experiments