Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods

Abstract:

A linear full elliptic second-order Partial Differential Equation (PDE), defined on a $d$-dimensional domain $\Omega$, is approximated by the isogeometric Galerkin method based on uniform tensor-product B-splines of degrees $(p_1,\ldots ,p_d)$. The considered approximation process leads to a $d$-level stiffness matrix, banded in a multilevel sense. This matrix is close to a $d$-level Toeplitz structure if the PDE coefficients are constant and the physical domain $\Omega$ is the hypercube $(0,1)^d$ without using any geometry map. In such a simplified case, a detailed spectral analysis of the stiffness matrices has already been carried out in a previous work. In this paper, we complete the picture by considering non-constant PDE coefficients and an arbitrary domain $\Omega$, parameterized with a non-trivial geometry map. We compute and study the spectral symbol of the related stiffness matrices. This symbol describes the asymptotic eigenvalue distribution when the fineness parameters tend to zero (so that the matrix-size tends to infinity). The mathematical tool used for computing the symbol is the theory of Generalized Locally Toeplitz (GLT) sequences.