@TechReport{ it:2016-001,
author = {Marco Donatelli and Ali Dorostkar and Mariarosa Mazza and
Maya Neytcheva and Stefano Serra-Capizzano},
title = {A Block Multigrid Strategy for Two-Dimensional Coupled
{PDE}s},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2016},
number = {2016-001},
month = jan,
abstract = {We consider the solution of linear systems of equations,
arising from the finite element approximation of coupled
differential boundary value problems. Letting the fineness
parameter tend to zero gives rise to a sequence of large
scale structured two-by-two block matrices. We are
interested in the efficient iterative solution of the so
arising linear systems, aiming at constructing optimal
preconditioning methods that are robust with respect to the
relevant parameters of the problem. We consider the case
when the originating systems are solved by a preconditioned
Krylov method, as inner solver, and propose an efficient
preconditioner for that, based on the Generalized Locally
Toeplitz framework.
In this paper, we exploit the almost two-level block
Toeplitz structure of the arising block matrix. We provide
a spectral analysis of the underlying matrices and then, by
exploiting the spectral information, we design a multigrid
method with an ad hoc grid transfer operator. As shown in
the included examples, choosing the damped Jacobi or
Gauss-Seidel methods as smoothers and using the resulting
solver as a preconditioner leads to a competitive strategy
that outperforms some aggregation-based algebraic multigrid
methodss, widely employed in the relevant literature.}
}