Analysis of the spectral symbol associated to discretization schemes of linear self-adjoint differential operators

Abstract

Given a linear self-adjoint differential operator mathscr {L} along with a discretization scheme (like Finite Differences, Finite Elements, Galerkin Isogeometric Analysis, etc.), in many numerical applications it is crucial to understand how good the (relative) approximation of the whole spectrum of the discretized operator mathscr {L},^{(n)} is, compared to the spectrum of the continuous operator mathscr {L}. The theory of Generalized Locally Toeplitz sequences allows to compute the spectral symbol function omega associated to the discrete matrix mathscr {L},^{(n)}. Inspired by a recent work by T. J. R. Hughes and coauthors, we prove that the symbol omega can measure, asymptotically, the maximum spectral relative error mathscr {E}ge 0. It measures how the scheme is far from a good relative approximation of the whole spectrum of mathscr {L}, and it suggests a suitable (possibly non-uniform) grid such that, if coupled to an increasing refinement of the order of accuracy of the scheme, guarantees mathscr {E}=0.