Guaranteed lower bounds of the smallest eigenvalues of elliptic differential operators

Abstract

This paper presents a rigorous analysis of the method of computing guaranteed lower bounds of the smallest eigenvalue of an elliptic operator in the case of mixed or purely Neumann boundary conditions. The method was originally invented in [8]. It is based on a decomposition of a domain into a set of overlapping subdomains, for which the corresponding estimates of minimal positive eigenvalues are known or easily computable.We prove that finding a guaranteed lower bound can be reduced to a finite dimensional variational problem. The dimension of the problem depends on the amount of subdomains, and the structure of the corresponding functional depends on topological properties of the set of overlapping subdomains. Several examples show the performance of the estimates.