Spectral distribution of PDE discretization matrices from isogeometric analysis: the case of $L^1$ coefficients and non-regular geometry

Abstract

We consider the matrices arising from the Galerkin B-spline Isogeometric Analysis (IgA) approximation of a $d$-dimensional second-order Partial Differential Equation (PDE). We compute the singular value and eigenvalue distribution of these matrices under minimal assumptions on the PDE coefficients and the geometry map involved in the IgA discretization. In particular, $L^1$ coefficients and non-regular geometries are allowed. The mathematical technique used in our derivation is entirely based on the theory of Generalized Locally Toeplitz (GLT) sequences, which is a quite general technique that can also be applied to several other PDE discretization methods.