Flux reconstructions in the Lehmann-Goerisch method for lower bounds on eigenvalues
This work improves computational efficiency and scalability for obtaining lower bounds on eigenvalues of elliptic operators, which is important for numerical verification in scientific computing.
The authors propose a simpler global problem for computing fluxes in the Lehmann-Goerisch method, which is smaller, positive definite, and allows for parallelization via local patch problems. Numerical examples show that local flux reconstructions enable computing lower bounds on eigenvalues on finer meshes than traditional global reconstructions.
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 $σ_i$ that approximate co-gradients of exact eigenfunctions scaled by corresponding eigenvalues. Fluxes $σ_i$ 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 $σ_i$ of the same quality. The simplified problem is smaller, it is positive definite, and any $H(\mathrm{div},Ω)$ 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.