Multiscale Discretization Scheme Based on the Rayleigh Quotient Iterative Method for the Steklov Eigenvalue Problem

Hai Bi,Yidu Yang

doi:10.1155/2012/487207

Abstract

This paper discusses efficient numerical methods for the Steklov eigenvalue problem and establishes a new multiscale discretization scheme and an adaptive algorithm based on the Rayleigh quotient iterative method. The efficiency of these schemes is analyzed theoretically, and the constants appeared in the error estimates are also analyzed elaborately. Finally, numerical experiments are provided to support the theory.

Highlights

Steklov eigenvalue problems have several deep applications both in physical and mechanical fields
A two-grid finite element discretization scheme is considered one of these efficient methods. This discretization technique was first introduced by Xu 13, for nonsymmetric and nonlinear elliptic problems, and due to its outstanding performance in computation, it has been successfully applied and further investigated for many other problems, for example, Poisson eigenvalue equations and integral equations in, nonlinear eigenvalue problems in, Schrodinger equation in 17, Stokes equations in, and so forth
This paper aims to establish a multi-scale discretization scheme based on the shifted-inverse power method, and an adaptive algorithm for the Steklov eigenvalue problem

Summary

Introduction

Steklov eigenvalue problems have several deep applications both in physical and mechanical fields. A two-grid finite element discretization scheme is considered one of these efficient methods. This discretization technique was first introduced by Xu 13, for nonsymmetric and nonlinear elliptic problems, and due to its outstanding performance in computation, it has been successfully applied and further investigated for many other problems, for example, Poisson eigenvalue equations and integral equations in , nonlinear eigenvalue problems in , Schrodinger equation in 17, , Stokes equations in , and so forth. This paper aims to establish a multi-scale discretization scheme based on the shifted-inverse power method, and an adaptive algorithm for the Steklov eigenvalue problem. Numerical experiments are provided to support our theoretical analysis

Preliminaries

Multi-Scale Discretization Scheme

Numerical Experiments

Concluding Remarks