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

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