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

2022/11/06 by Xuefeng Liu, Liu, Xuefeng, Tomáš Vejchodský +1
Engineering · Physics and Astronomy · #65N25 #65N30 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2211.03218

openalex publication_date 2022/11/06 · openalex created_date 2022/11/13 · openalex updated_date 2026/07/28

Abstract

For conforming finite element approximations of the Laplacian eigenfunctions, a fully computable guaranteed error bound in the L² 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 .

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