Towards Robust Algebraic Multigrid Methods for Nonsymmetric Problems

This thesis presents a rigorous, abstract analysis of multigrid methods for positive nonsymmetric problems, particularly suited to algebraic multigrid, with a completely new approach to nonsymmetry which is based on a new concept of absolute value for nonsymmetric operators.
Multigrid, and in particular algebraic multigrid, has become an indispensable tool for the solution of discretizations of partial differential equations. While used in both the symmetric and nonsymmetric cases, the theory for the nonsymmetric case has lagged substantially behind that for the symmetric case. This thesis closes some of this gap, presenting a major and highly original contribution to an important problem of computational science.
The new approach to nonsymmetry will be of interest to anyone working on the analysis of discretizations of nonsymmetric operators, even outside the context of multigrid. The presentation of the convergence theory may interest even those only concerned with the symmetric case, as it sheds some new light on and extends existing results.

Nominated as an outstanding Ph.D. thesis by the University of Oxford, UK Provides a concise and self-contained introduction to multigrid for a simple model problem Presents a new absolute value concept that naturally extends the energy norm to the nonsymmetric case Presents a novel general convergence theory for two-level methods, the first to treat interpolation and restriction independently, by making use of the new absolute value Includes supplementary material: sn.pub/extras

Inhalt
Introduction.- Theoretical Foundations.- Form Absolute Value.- Convergence Theory.- Application to a New AMG Method.- Conclusions.