Spectral analysis of coupled PDEs and of their Schur complements via Generalized Locally Toeplitz sequences in 2D

摘要

We consider large linear systems of algebraic equations arising from the Finite Element approximation of coupled partial differential equations. As case study we focus on the linear elasticity equations, formulated as a saddle point problem to allow for modelling of purely incompressible materials. To analyse the properties of the arising matrices we use the notion of the so-called spectral symbol in the Generalized Locally Toeplitz (GLT) setting. The fruitful idea behind GLT is that it allows to associate a function, a symbol, with instances of a broad class of structured matrices. The symbol describes the spectrum of the corresponding matrix, easing in this way its analysis and guiding the construction of efficient approximations of the matrix to be used as preconditioners or solvers in a multigrid context. We derive the GLT symbol of the sequence of matrices {A(n)} approximating the elasticity equations. Further, exploiting the property that the GLT class defines an algebra of matrix sequences and the fact that the Schur complements are obtained via elementary algebraic operation on the blocks of A(n), we derive the symbols f(S) of the associated sequences of Schur complements {S-n}. As a consequence of the GLT theory, the eigenvalues of S-n for large n are described by a sampling of f(S) on a uniform grid of its domain of definition. We extend the existing GLT technique with novel elements, related to block-matrices and Schur complement matrices, and illustrate the theoretical findings with numerical tests. (C) 2016 Elsevier B.V. All rights reserved.