Domain Decomposition Methods and Preconditioning

Abstract Domain decomposition (DD) methods nowadays provide powerful tools for constructing efficient parallel solvers for large‐scale systems of algebraic equations arising from the discretization of partial differential equations. The classical alternating Schwarz method and the classical substructuring technique have led to advanced overlapping and nonoverlapping domain decomposition solvers (preconditioners), which can be analyzed from a unified point of view now calledSchwarz theory. This survey chapter starts with a brief historical overview, provides the basic results of the Schwarz theory, looks at some overlapping domain decomposition methods (preconditioners) in brief, and discusses more extensively various nonoverlapping domain decomposition techniques. Some recent advances in the development of fast DD algorithms for finite element and isogeometric analysis discretizations of elliptic problems are also described.