Lusin Theorem, GLT Sequences and Matrix Computations: An Application to the Spectral Analysis of PDE Discretization Matrices

Carlo Garoni, Carla Manni, Stefano Serra-Capizzano, Debora Sesana, and Hendrik Speleers

March 2015

Abstract:

We extend previous results on the spectral distribution of discretization matrices arising from B-spline Isogeometric Analysis (IgA) approximations of a general d-dimensional second-order elliptic Partial Differential Equation (PDE) with variable coefficients.

First, we provide the spectral symbol of the Galerkin B-spline IgA stiffness matrices, assuming only that the PDE coefficients 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 (theta₁,...,theta_d), 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 be also applied to other PDE discretization techniques than B-spline IgA.

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