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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 projectionof the conforming finite element method, and has an optimal speed ofconvergence for the eigenfunctions with the worst regularity. The resultingerror estimate bounds the distance of spaces of exact and approximateeigenfunctions and, hence, is robust even in the case of multiple and tightlyclustered eigenvalues. The accuracy of the proposed bound is illustrated bynumerical examples. The demonstration code is available athttps://ganjin.online/xfliu/EigenfunctionEstimation4FEM .

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