A Cascadic Multigrid Method for Eigenvalue Problem

TL;DR

This paper introduces a cascadic multigrid method for eigenvalue problems that efficiently combines smoothing steps and coarse space solutions to achieve optimal convergence and computational efficiency.

Contribution

It proposes a novel multilevel correction scheme that improves the efficiency of solving eigenvalue problems using multigrid techniques.

Findings

Achieves optimal convergence rate

Reduces computational work compared to traditional methods

Validated by numerical experiments

Abstract

A cascadic multigrid method is proposed for eigenvalue problems based on the multilevel correction scheme. With this new scheme, an eigenvalue problem on the finest space can be solved by smoothing steps on a series of multilevel finite element spaces and eigenvalue problem solving on the coarsest finite element space. Choosing the appropriate sequence of finite element spaces and the number of smoothing steps, the optimal convergence rate with the optimal computational work can be arrived. Some numerical experiments are presented to validate our theoretical analysis.