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Preconditioning the Poincaré-Steklov operator by using Green's function
Author(s):
Jinchao
Xu;
Sheng
Zhang.
Journal:
Math. Comp.
66
(1997),
125-138.
MSC (1991):
Primary 65N20, 65F10
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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.
References:
- 1.
- J. H. Bramble, J. E. Pasciak, and A. H. Schatz. The construction of preconditioners for elliptic problems by substructuring, I. Math. Comp., 47:103-134, 1986. MR 87m:65174
- 2.
-
-, An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp., 46:361-369, 1986. MR 88a:65123 - 3.
-
-, The construction of preconditioners for elliptic problems by substructuring, II. Math. Comp., 49:1-16, 1987. MR 88j:65248 - 4.
-
-, The construction of preconditioners for elliptic problems by substructuring, IV. Math. Comp., 53:1-24, 1989. MR 89m:65098 - 5.
- J. H. Bramble, J. E. Pasciak, and J. Xu. A multilevel preconditioner for domain decomposition boundary systems. In Proceeding of 10th International Conference on Computing Methods in Applied Sciences and Engineering. INRIA, 1991.
- 6.
- M. Dryja. A method of domain decomposition for 3-D finite element elliptic problems. In R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, editors, First International Symposium on Domain Decomposition Methods for Partial Differential Equations, Philadelphia, PA, 1988. SIAM. MR 90b:65200
- 7.
- M. Dryja and O. B. Widlund. Domain decomposition algorithms with small overlap. SIAM J. Sci. Statist. Comput., 15:604-620, 1994. MR 95d:65102
- 8.
- P. Grisvard. Elliptic problems in nonsmooth domains. Pitman Advanced Publishing Program, Boston, 1985. MR 86m:35044
- 9.
- L. Greengard J. Carrier and V. Rokhlin. A fast adaptive multipole algorithm for particle simulations. SIAM J. Sci. Statist. Comput., 9:669-686, 1988. MR 89e:65008
- 10.
- J. L. Lions and E. Magenes. Nonhomogeneous Boundary Value Problems and Applications, volume I. Springer, New York, Heidelberg, Berlin, 1972. MR 50:2670
- 11.
- A. M. Matsokin and S. V. Nepomnyashchikh. A Schwarz alternating method in a subspace. Soviet Mathematics, 29:78-84, 1985.
- 12.
-
-, Norms in the space of traces of mesh functions. Soviet J. Numer. Anal. Math. Modelling, 3:199-216, 1988. - 13.
- J. Ne[??]cas. Les Méthodes directes en Théorie des Équations Elliptiques. Academia, Prague, 1967. MR 37:3168
- 14.
- B. F. Smith. An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems. SIAM J. Sci. Statist. Comput., 13:364-378, 1992. MR 92g:73080
- 15.
- B. F. Smith and O. B. Widlund. A domain decomposition algorithm using a hierarchical basis. SIAM J. Sci. Statist. Comput., 11:1212-1220, 1990. MR 91m:65125
- 16.
- C. H. Tong, T. F. Chan and C. J. Kuo. A domain decomposition preconditioner based on a change to a multilevel nodal basis. SIAM J. Sci. Statist. Comput., 12:1486-1495, 1991. MR 92i:65070
- 17.
- J. Xu. Theory of multilevel methods. Ph.D. thesis, Cornell University, May 1989.
- 18.
-
-, Iterative methods by space decomposition and subspace correction. SIAM Review, 34:581-613, 1992. MR 93k:65029 - 19.
-
-, Multigrid and domain decomposition methods. Research Monograph (preprint), 1995.
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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:
10.1090/S0025-5718-97-00799-0
PII:
S 0025-5718(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.
Copyright of article:
Copyright
1997,
American Mathematical Society
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