Lusin theorem, GLT sequences and matrix computations: An application to the spectral analysis of PDE discretization matrices

Abstract

In this paper we consider a general d-dimensional second-order elliptic Partial Differential Equation (PDE) with variable coefficients, and we extend previous results on the spectral distribution of discretization matrices arising from B-spline Isogeometric Analysis (IgA). First, we provide the spectral symbol of the Galerkin B-spline IgA stiffness matrices, under the assumption that the PDE coefficients only belong to L∞. This symbol describes the asymptotic spectral distribution when the fineness parameters tend to zero (so that the matrix-size tends to infinity). Second, we prove the positive semi-definiteness of the d×d symmetric matrix in the Fourier variables (θ1,…,θd), which appears in the expression of the symbol. This matrix is related to the discretization of the (negative) Hessian operator, and its positive semi-definiteness implies the non-negativity of the symbol. The mathematical arguments used in our derivation are based on the Lusin theorem, on the theory of Generalized Locally Toeplitz (GLT) sequences, and on careful Linear Algebra manipulations of matrix determinants. These arguments are very general and can also be applied to other (local) PDE discretization methods, different from B-spline IgA.