Preprint 6-2007

Theoretical and Numerical Comparison of Various Projection Methods derived from Deflation, Domain Decomposition and Multigrid Methods (extended version)

Author(s) : J. M. Tang , R. Nabben , K. Vuik , Y. A. Erlangga

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 6-2007

ZDM : N

Abstract :
For various applications, it is well-known that a two-level-preconditioned Krylov method is an efficient method for solving large and sparse linear systems. Beside a traditional preconditioner like incomplete Cholesky decomposition, a projector has been included as preconditioner to get rid of a number of small and large eigenvalues of the matrix. In literature, various projection methods are known coming from the fields of deflation, domain decomposition and multigrid. From an abstract point of view, these methods are closely related. The aim of this paper is to compare these projection methods both theoretically and numerically using various elliptic test problems. We investigate their convergence properties and stability by considering implementation issues, rounding-errors, inexact coarse solves and severe termination criteria. Finally, we end up with a suggestion of the optimal second-level preconditioner, which is as stable as the abstract balancing preconditioner and as cheap and fast as the deflation preconditioner.

Keywords : deflation, domain decomposition, multigrid, preconditioning, Krylov methods, implementation, Poisson equation, hybrid methods, coarse grid corrections