Spectral behavior of matrix sequences and discretized boundary value problems

In this paper we provide theoretical tools for dealing with the spectral properties of general sequences of matrices of increasing dimension. More specifically, we give a unified treatment of notions such as distribution, equal distribution, localization, equal localization, clustering and sub-clustering. As a case study we consider the matrix sequences arising from the finite difference (FD) discretization of elliptic and semielliptic boundary value problems (BVPs). The spectral analysis is then extended to Toeplitz-based preconditioned matrix sequences with special attention to the case where the coefficients of the differential operator are not regular (belong to L1) and to the case of multidimensional problems. The related clustering properties allow the establishment of some ergodic formulas for the eigenvalues of the preconditioned matrices.