Algebraic Multigrid for Convection-dominated Problems
In many fields of our research there is a desperate need for fast and
efficient solvers. Multigrid Methods are among the best methods for solving
symmetric positive definite systems. However, for many applications (e.g.
Computational Fluid Dynamics) convective phenomena play an
important role and may dominate the whole process. Convective terms result in
nonsymmetric linear systems that often turn out to be very challenging for
iterative solving methods. We develop a new framework for algebraic multigrid
preconditioners that is also appropriate for these kind of nonsymmetric
systems. Furthermore it provides flexibility that allows the continued
development of robust AMG solvers.
Algebraic multigrid methods use only the system matrix for generating
inter-level multigrid transfer operators. Inspired by Petrov-Galerkin
smoothed aggregation methods we use different Galerkin projections for the
prolongation and restriction operators. The key idea for our transfer
operators is connected to approximating a Schur complement. With special
Galerkin projections we preserve important low-frequency error components and
limit the sparsity pattern of the multigrid transfer operators.
In numerical experiments we were able to demonstrate that the resulting methods are competitive with state-of-the-art multigrid transfer strategies and even perform noticably better in many situations.