Projection error-based guaranteed L2 error bounds for finite element approximations of Laplace eigenfunctions

TL;DR

This paper introduces a fully computable, guaranteed L2 error bound for finite element approximations of Laplace eigenfunctions, ensuring robustness even with multiple or clustered eigenvalues.

Contribution

It presents a novel, guaranteed error bound based on a priori estimates that is optimal for eigenfunctions with varying regularity, improving reliability of finite element methods.

Findings

The error bound is fully computable and guaranteed.

The bound is robust for multiple and clustered eigenvalues.

Numerical examples confirm the accuracy of the error estimate.

Abstract

For conforming finite element approximations of the Laplacian eigenfunctions, a fully computable guaranteed error bound in the $L^2$ norm sense is proposed. The bound is based on the a priori error estimate for the Galerkin projection of the conforming finite element method, and has an optimal speed of convergence for the eigenfunctions with the worst regularity. The resulting error estimate bounds the distance of spaces of exact and approximate eigenfunctions and, hence, is robust even in the case of multiple and tightly clustered eigenvalues. The accuracy of the proposed bound is illustrated by numerical examples. The demonstration code is available at https://ganjin.online/xfliu/EigenfunctionEstimation4FEM .