Preconditioning the Poincaré-Steklov operator by using Green's function

Jinchao Xu, Sheng Zhang

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

This paper is concerned with the Poincaré-Steklov operator that is widely used in domain decomposition methods. It is proved that the inverse of the Poincaré-Steklov operator can be expressed explicitly by an integral operator with a kernel being the Green's function restricted to the interface. As an application, for the discrete Poincaré-Steklov operator with respect to either a line (edge) or a star-shaped web associated with a single vertex point, a preconditioner can be constructed by first imbedding the line as the diameter of a disk, or the web as a union of radii of a disk, and then using the Green's function on the disk. The proposed technique can be effectively used in conjunction with various existing domain decomposition techniques, especially with the methods based on vertex spaces (from multi-subdomain decomposition). Some numerical results are reported.

Original languageEnglish (US)
Pages (from-to)125-138
Number of pages14
JournalMathematics of Computation
Volume66
Issue number217
StatePublished - Jan 1 1997

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Preconditioning
Green's function
Mathematical operators
Decomposition
Domain decomposition methods
Operator
Stars
Imbedding
Line
Decomposition Techniques
Domain Decomposition Method
Domain Decomposition
Vertex of a graph
Integral Operator
Preconditioner
Star
Union
Radius
kernel
Decompose

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

Cite this

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abstract = "This paper is concerned with the Poincar{\'e}-Steklov operator that is widely used in domain decomposition methods. It is proved that the inverse of the Poincar{\'e}-Steklov operator can be expressed explicitly by an integral operator with a kernel being the Green's function restricted to the interface. As an application, for the discrete Poincar{\'e}-Steklov operator with respect to either a line (edge) or a star-shaped web associated with a single vertex point, a preconditioner can be constructed by first imbedding the line as the diameter of a disk, or the web as a union of radii of a disk, and then using the Green's function on the disk. The proposed technique can be effectively used in conjunction with various existing domain decomposition techniques, especially with the methods based on vertex spaces (from multi-subdomain decomposition). Some numerical results are reported.",
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Preconditioning the Poincaré-Steklov operator by using Green's function. / Xu, Jinchao; Zhang, Sheng.

In: Mathematics of Computation, Vol. 66, No. 217, 01.01.1997, p. 125-138.

Research output: Contribution to journalArticle

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