Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem

Liu Chen,Qiumei Huang,Fei Xu

doi:10.1051/m2an/2021076

Abstract

In this paper, we propose a local defect-correction method for solving the Steklov eigenvalue problem arising from the scalar second order positive definite partial differential equations based on the multilevel discretization. The objective is to avoid solving large-scale equations especially the large-scale Steklov eigenvalue problem whose computational cost increases exponentially. The proposed algorithm transforms the Steklov eigenvalue problem into a series of linear boundary value problems, which are defined in a multigrid space sequence, and a series of small-scale Steklov eigenvalue problems in a coarse correction space. Furthermore, we use the local defect-correction technique to divide the large-scale boundary value problems into small-scale subproblems. Through our proposed algorithm, we avoid solving large-scale Steklov eigenvalue problems. As a result, our proposed algorithm demonstrates significantly improved the solving efficiency. Additionally, we conduct numerical experiments and a rigorous theoretical analysis to verify the effectiveness of our proposed approach.

Highlights

Owing to continuous advancements in computer technology and computing technology, computational science and engineering has become the third approach for conducting scientific and engineering research after experimentation and theoretical analysis
Local defect-correction method based on multilevel discretization for solving the Steklov eigenvalue problem
We introduce the local defect-correction method based on multilevel discretization for solving the Steklov eigenvalue problem

Summary

Introduction

Owing to continuous advancements in computer technology and computing technology, computational science and engineering has become the third approach for conducting scientific and engineering research after experimentation and theoretical analysis. The development of local defect-correction methods (or local and parallel methods) has progressed rapidly because its use is significantly convenient in large-scale scientific and engineering computing This computational technique for solving linear elliptic equations was first proposed by Xu and Zhou [51]. We design an efficient local defect-correction method for solving the Steklov eigenvalue problems from the scalar second order positive definite partial differential equations. Because the main computational work of this algorithm is controlled by the linear boundary value problem, which can be solved efficiently using the local defect-correction technique, its efficiency in solving the Steklov eigenvalue problem can be significantly improved.

Finite element method for solving the elliptic boundary value problem

Finite element method for solving the Steklov eigenvalue problem

One step of the local defect-correction method

Local defect-correction method based on multilevel discretization

Estimate of computational work

Example 1

Example 2

Findings

Concluding remark