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

TL;DR

This paper introduces a simplified, positive definite flux reconstruction method for the Lehmann-Goerisch eigenvalue bounds, enabling efficient parallel computation and finer mesh analysis.

Contribution

A new simplified global flux reconstruction approach that is positive definite, easier to compute, and allows for parallelization and finer mesh analysis.

Findings

The new method produces fluxes of the same quality as traditional approaches.

It enables parallel computation through local problem splitting.

Numerical examples demonstrate improved bounds on finer meshes.

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 $\sigma_i$ that approximate co-gradients of exact eigenfunctions scaled by corresponding eigenvalues. Fluxes $\sigma_i$ are usually computed by a global saddle point problem solved by mixed finite element methods. In this paper we propose a simpler global problem that yields fluxes $\sigma_i$ of the same quality. The simplified problem is smaller, it is positive 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 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 to compute lower bounds on eigenvalues on considerably finer meshes than the traditional global reconstructions.