Multigrid solution techniques for anisotropic structured linear systems Original Research Article

Multigrid methods are highly efficient solution techniques for large sparse structured linear systems which are positive definite and ill-conditioned. In a recent paper [R. Fischer, T. Huckle, Multigrid methods for anisotropic BTTB systems, Linear Algebra Appl. (2005), submitted for publication], multigrid methods have been developed which are especially designed for anisotropic matrices belonging to the two-level Toeplitz class. These methods are primarily based on the use of a suitable combination of semicoarsening and full coarsening steps. In this paper the main focus is on the design of efficient smoothing techniques. Moreover, we are not only interested in two-level Toeplitz matrices, but also in matrices of two-level trigonometric matrix algebras. First, we describe methods for systems with anisotropy along coordinate axes. Although some of the ideas are known from the solution of partial differential equations, we present them here in a more formal way using generating functions and their level curves. This allows us not only to obtain theoretical results on convergence and reduction of anisotropy, but also to carry over the results to systems with anisotropy in other directions. We introduce new coordinates in order to describe these more complicated systems in terms of generating functions. This enables us to develop smoothers which are especially suitable for these more complicated systems.