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We derive optimal and asymptotically exact a posteriori error estimates forthe approximation of the Laplace eigenvalue problem. To do so, we combine tworesults from the literature. First, we use the hypercircle techniques developedfor mixed eigenvalue approximations with Raviart-Thomas Finite elements. Inaddition, we use the post-processings introduced for the eigenvalue andeigenfunction based on mixed approximations with the Brezzi-Douglas-MariniFinite element. To combine these approaches, we define a novel additional localpost-processing for the fluxes that appropriately modifies the divergence.Consequently, the new flux can be used to derive upper bounds and still showsgood approximation properties. Numerical examples validate the theory andmotivate the use of an adaptive mesh refinement.

A note on asymptotically exact a posteriori error estimates for mixed Laplace eigenvalue problems