  We derive optimal and asymptotically exact a posteriori error estimates for
the approximation of the Laplace eigenvalue problem. To do so, we combine two
results from the literature. First, we use the hypercircle techniques developed
for mixed eigenvalue approximations with Raviart-Thomas Finite elements. In
addition, we use the post-processings introduced for the eigenvalue and
eigenfunction based on mixed approximations with the Brezzi-Douglas-Marini
Finite element. To combine these approaches, we define a novel additional local
post-processing for the fluxes that appropriately modifies the divergence.
Consequently, the new flux can be used to derive upper bounds and still shows
good approximation properties. Numerical examples validate the theory and
motivate the use of an adaptive mesh refinement.
