Domain Decomposition Methods and Preconditioning

Abstract

Domain decomposition 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), that can be analyzed from a unified point of view now called Schwarz theory . This survey article 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.

A Nitsche-type formulation and comparison of the most common domain decomposition methods in isogeometric analysis

“…These methods can be applied in problems that involve different physics at each subdomain, for example, fluid-structure interaction [15][16][17][18][19] and in problems comprising several subdomains with the same underlying physics, for example, contact mechanics [20,21]. Another field of application of the DDMs is parallel computing [22], where the domain is decomposed out of computational reasons and not because of problem setting itself. Generally, the DDMs can be distinguished into overlapping and non-overlapping.…”

Iterated Ritz procedure for solving linear algebraic equations systems

“…they only contribute to proper understanding and emphasize the generality of this iterative algorithm. It should also be mentioned that very popular single or multiple preconditioning [13,14] can be interpreted using (one or several) coordinate vectors. For this procedure, the preconditioning does not imply any significant change of the algorithm (cf.…”

“…it is sufficient to differentiate the energy increase (13) If the generalized Ritz stiffness matrix is introduced (14) which is also symmetric, and in the case of linearly independent coordinate vectors also positive definite, and if the generalized Ritz load vector is defined as (15) the energy increase can be written in an abbreviated form (16) to which after minimization (differentiation) according to , the following equation system can be related (17) In the sense of the Ritz method, this system is used to approximate the initial one (1). As the approximation is usually unsatisfactory, an iterative improvement should be made.…”

A robust time‐space formulation for large‐scale scalar wave problems using exponential basis functions

“…The use of overlapping subdomains helps us to evade using a sophisticated mesh generation algorithm as the preprocessor. In this regard, here, we follow the multiplicative Schwarz algorithm, which is an iterative one (see the work of Korneev and Langer for more details). The formulation starts with assuming that Ω d in is formed by the union of a set of subdomains.…”