Convergence analysis of two-grid methods for nonsymmetric positive definite systems
It advances the theoretical foundation of multigrid methods for nonsymmetric systems, which is important for numerical linear algebra but is an incremental step in a specialized domain.
This paper provides a convergence analysis of two-grid methods for nonsymmetric positive definite systems, deriving an identity for the convergence factor with exact coarse-grid solves and a theory for inexact solves under mild assumptions.
The convergence theory of multigrid methods for symmetric positive definite systems is well established. For nonsymmetric systems, however, the corresponding theory remains far from mature. Two-grid analysis is fundamental to the design and analysis of multigrid methods. This paper presents a convergence analysis of two-grid methods for nonsymmetric positive definite systems. When the coarse-grid system is solved exactly, we derive a succinct identity for the two-grid convergence factor measured in a smoother-induced norm. More generally, under mild assumptions, we develop a convergence theory for inexact two-grid methods, where convergence is measured in a generic norm.