Fully computable a posteriori error bounds for eigenfunctions

Guaranteed a posteriori estimates on the error of approximate eigenfunctions in both energy and $L^2$ norms are derived for the Laplace eigenvalue problem. The problem of ill-conditioning of eigenfunctions in case of tight clusters and multiple eigenvalues is solved by estimating the directed distance between the spaces of exact and approximate eigenfunctions. The error estimates for approximate eigenfunctions are based on rigorous lower and upper bounds on eigenvalues. Such eigenvalue bounds can be computed for example by the finite element method along with the recently developed explicit error estimation [24] and the Lehmann--Goerisch method. The efficiency of the derived error bounds for eigenfunctions is illustrated by numerical examples.

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