A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems

Xiao‐Chuan ‐C Cai*, William D. Gropp, David E. Keyes

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


In recent years, competitive domain‐decomposed preconditioned iterative techniques of Krylov‐Schwarz type have been developed for nonsymmetric linear elliptic systems. Such systems arise when convection‐diffusion‐reaction problems from computational fluid dynamics or heat and mass transfer are linearized for iterative solution. Through domain decomposition, a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow effective solution on parallel machines. A central question is how to choose these small problems and how to arrange the order of their solution. Different specifications of decomposition and solution order lead to a plethora of algorithms possessing complementary advantages and disadvantages. In this report we compare several methods, including the additive Schwarz algorithm, the classical multiplicative Schwarz algorithm, an accelerated multiplicative Schwarz algorithm, the tile algorithm, the CGK algorithm, the CSPD algorithm, and also the popular global ILU‐family of preconditioners, on some nonsymmetric or indefinite elliptic model problems discretized by finite difference methods. The preconditioned problems are solved by the unrestarted GMRES method. A version of the accelerated multiplicative Schwarz method is a consistently good performer.

Original languageEnglish (US)
Pages (from-to)477-504
Number of pages28
JournalNumerical Linear Algebra with Applications
Issue number5
StatePublished - 1994
Externally publishedYes


  • Domain decomposition
  • Iterative methods
  • Nonsymmetric and/or indefinite elliptic problems
  • Preconditioning

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics


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