Generalized spectrum of second order differential operators: 3D problems

Abstract

Generalized spectra of differential operators can be related to spectra of preconditioned discretized operators. Obtaining (estimates of) the eigenvalues of the preconditioned discretized operators may lead to better estimating of the quality of preconditioners. In this short paper, we answer the open question posted in the recent paper Generalized spectrum of second order differential operators, authored by Gergelits, Nielsen, and Strako\v s. The proof we present allows us to fully extend characterizing the generalized spectra of $\nabla\cdot K\nabla u=\lambda\triangle u$ to problems of dimension three or higher.