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The standard application of the Lehmann-Goerisch method for lower bounds oneigenvalues of symmetric elliptic second-order partial differential operatorsrelies on determination of fluxes $\sigma_i$ that approximate co-gradients ofexact eigenfunctions scaled by corresponding eigenvalues. Fluxes $\sigma_i$ areusually computed by a global saddle point problem solved by mixed finiteelement methods. In this paper we propose a simpler global problem that yieldsfluxes $\sigma_i$ of the same quality. The simplified problem is smaller, it ispositive definite, and any $H(\mathrm{div},\Omega)$ conforming finite elements,such as Raviart-Thomas elements, can be used for its solution. In addition,these global problems can be split into a number of independent local problemson patches, which allows for trivial parallelization. The computationalperformance of these approaches is illustrated by numerical examples forLaplace and Steklov type eigenvalue problems. These examples also show thatlocal flux reconstructions enable to compute lower bounds on eigenvalues onconsiderably finer meshes than the traditional global reconstructions.

Flux reconstructions in the Lehmann-Goerisch method for lower bounds on eigenvalues