An Algebraic Multigrid Method for Eigenvalue Problems in Some Different Cases

TL;DR

This paper introduces an algebraic multigrid method combined with a multilevel correction scheme to efficiently solve large-scale eigenvalue problems, especially on complex, unstructured meshes.

Contribution

It develops a novel algebraic multigrid approach for eigenvalue problems that is efficient, flexible, and converges globally regardless of the number of eigenvalues.

Findings

Efficient solution of large-scale eigenvalue problems on unstructured meshes.

Global convergence independent of the number of eigenvalues.

Applicable to problems with domain singularities and discontinuous parameters.

Abstract

The aim of this paper is to develop an algebraic multigrid method to solve eigenvalue problems based on the combination of the multilevel correction scheme and the algebraic multigrid method for linear equations. Our approach uses the algebraic multigrid method setup procedure to construct the hierarchy and the intergrid transfer operators. In this algebraic multigrid scheme, a large scale eigenvalue problem is solved by some algebraic multigrid smoothing steps in the hierarchy and very small-dimensional eigenvalue problems. To emphasize the efficiency and flexibility of the proposed method, here we consider a set of test eigenvalue problems, discretized on unstructured meshes, with different shape of domain, singularity, and discontinuous parameters. Moreover, global convergence independent of the number of desired eigenvalues is obtained.