Analysis of the spectral symbol function for spectral approximation of a differential operator
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Given a differential operator L along with its own eigenvalue problem Lu = λ u and an associated algebraic equation L^(n)u_n = λu_n obtained by means of a discretization scheme (like Finite Differences, Finite Elements, Galerkin Isogeometric Analysis, etc.), the theory of Generalized Locally Toeplitz (GLT) sequences serves the purpose to compute the spectral symbol function ω associated to the discrete operator L^(n) We prove that the spectral symbol ω provides a necessary condition for a discretization scheme in order to uniformly approximate the spectrum of the original differential operator L. The condition measures how far the method is from a uniform relative approximation of the spectrum of L. Moreover, the condition seems to become sufficient if the discretization method is paired with a suitable (non-uniform) grid and an increasing refinement of the order of approximation of the method. On the other hand, despite the numerical experiments in many recent literature, we disprove that in general a uniform sampling of the spectral symbol ω can provide an accurate relative approximation of the spectrum, neither of L nor of the discrete operator L^(n).
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