Preconditioning the Poincaré-Steklov operator

by using Green's function

Authors:
Jinchao Xu and Sheng Zhang

Journal:
Math. Comp. **66** (1997), 125-138

MSC (1991):
Primary 65N20, 65F10

DOI:
https://doi.org/10.1090/S0025-5718-97-00799-0

MathSciNet review:
1372010

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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.

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Additional Information

**Jinchao Xu**

Affiliation:
Department of Mathematics, Penn State University. University Park, Pennsylvania 16802

Email:
xu@math.psu.edu

**Sheng Zhang**

Affiliation:
State Key Laboratory of Scientific and Engineering Computing, Computing Center, Chinese Academy of Sciences, Beijing 100080, P.R. China

Email:
zhang_s@math.psu.edu

DOI:
https://doi.org/10.1090/S0025-5718-97-00799-0

Keywords:
Domain decomposition,
preconditioner,
Schur complement,
Green's function,
multigrid,
Poincar\'e-Steklov operator

Received by editor(s):
May 10, 1995

Received by editor(s) in revised form:
July 31, 1995, and January 26, 1996

Additional Notes:
This work was partially supported by National Science Foundation, Chinese Academy of Sciences and China National Natural Science funds.

Article copyright:
© Copyright 1997
American Mathematical Society