Energy Error Estimates of Subspace Method and Multigrid Algorithm for Eigenvalue Problems

TL;DR

This paper introduces new energy norm error estimates for subspace methods in eigenvalue problems, relates them to $L^2$-norm errors, and develops a new inverse power method with convergence analysis, applying it to multigrid algorithms.

Contribution

It provides a novel energy norm error estimate for subspace methods and links it to $L^2$-norm errors, along with a new inverse power method and multigrid analysis.

Findings

New energy norm error estimates for subspace methods

Relation between energy norm and $L^2$-norm errors established

Convergence analysis of a new inverse power method and multigrid algorithms

Abstract

This paper is to give a new understanding and applications of the subspace projection method for selfadjoint eigenvalue problems. A new error estimate in the energy norm, which is induced by the stiff matrix, of the subspace projection method for eigenvalue problems is given. The relation between error estimates in $L^2$-norm and energy norm is also deduced. Based on this relation, a new type of inverse power method is designed for eigenvalue problems and the corresponding convergence analysis is also provided. Then we present the analysis of the geometric and algebraic multigrid methods for eigenvalue problems based on the convergence result of the new inverse power method.