Energy Error Estimates of Subspace Method and Multigrid Algorithm for Eigenvalue Problems
This work offers theoretical insights and improved convergence analysis for numerical eigenvalue solvers, but is incremental as it builds on existing subspace and multigrid methods.
The paper provides new energy norm error estimates for the subspace projection method for selfadjoint eigenvalue problems, derives a relation between L2 and energy norm errors, and designs a new inverse power method with convergence analysis, which is then used to analyze multigrid methods for eigenvalue problems.
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.