Spectral analysis of coupled PDEs and of their Schur complements via the notion of Generalized Locally Toeplitz sequences

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 modeling of purely incompressible materials. Using the notion of the so-called spectral symbol in the Generalized Locally Toeplitz (GLT) setting, we derive the GLT symbol (in the Weyl sense) of the sequence of matrices {An} 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 An, we derive the symbols f S of the associated sequences of Schur complements {Sn}. As a consequence of the GLT theory, the eigenvalues of Sn 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.