A block multigrid strategy for two-dimensional coupled PDEs

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 preconditioner leads to a competitive strategy that outperforms some aggregation-based algebraic multigrids, widely employed in the relevant literature.