Multilevel preconditioning in H(div) and applications to a posteriori error estimates for discontinuous Galerkin approximations

Priv.-Doz. Dr. Johannes Kraus

Nov. 11, 2008, 2:30 p.m. MZ 005A

In this talk we present an algebraic multilevel iterative (AMLI) method for solving linear systems arising from finite element discretization of certain minimization or boundary-value problems that have their weak formulation in the space H(div).

In particular we focus on an efficient solution of the discrete problem related to advanced (functional-type) a posteriori error estimates, see e.g., Repin (2003). As recently observed, when using these estimates for discontinuous Galerkin approximations of elliptic problems, the computation of such guaranteed a posteriori error estimates can be even more expensive than computing the solution of the original problem, and thus, demands fast iterative solvers. Preconditioners for such linear systems within the framework of domain decomposition and multigrid techniques have been studied by several authors, see e.g., Vassilevski and Wang (1992), Arnold, Falk and Winther (1997, 2000) and Hiptmair (1997). Our work is based on a different approach, namely, hierarchical basis multilevel methods.