A Full Multigrid Method for Eigenvalue Problems

TL;DR

This paper introduces a full multigrid scheme for eigenvalue problems that transforms the problem into a series of boundary value and eigenvalue problems on multilevel finite element spaces, enhancing computational efficiency.

Contribution

It presents a novel nested multigrid method that reduces eigenvalue problems to boundary value problems solved efficiently on multiple levels.

Findings

Achieves optimal computational order similar to source problem solving.

Improves efficiency of eigenvalue problem solutions.

Flexible use of various linear solvers within the multigrid framework.

Abstract

In this paper, a full (nested) multigrid scheme is proposed to solve eigenvalue problems. The idea here is to use the multilevel correction method to transform the solution of eigenvalue problem to a series of solutions of the corresponding boundary value problems and eigenvalue problems defined on the coarsest finite element space. The boundary value problems which are define on a sequence of multilevel finite element space can be solved by some multigrid iteration steps. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as linear problem solver. The computational work of this new scheme can reach optimal order the same as solving the corresponding source problem. Therefore, this type of iteration scheme improves the efficiency of eigenvalue problem solving.