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

Abstract

The standard application of the Lehmann–Goerisch method for lower bounds on eigenvalues of symmetric elliptic second-order partial differential operators relies on determination of fluxes ${\tilde{\mathit{\sigma }}}_i$ that approximate co-gradients of exact eigenfunctions scaled by corresponding eigenvalues. Fluxes ${\tilde{\mathit{\sigma }}}_i$ are usually computed by solving a global saddle point problem with mixed finite element methods. In this paper we propose a simpler global problem that yields fluxes ${\tilde{\mathit{\sigma }}}_i$ of the same quality. The simplified problem is smaller, it is positive definite, and any $\mathit{H}(div,\Omega )$ 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 problems on patches, which allows for trivial parallelization. The computational performance of these approaches is illustrated by numerical examples for Laplace and Steklov type eigenvalue problems. These examples also show that local flux reconstructions enable computation of lower bounds on eigenvalues on considerably finer meshes than the traditional global reconstructions.