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

Xu, Jinchao; Zhang, Sheng

doi:10.1090/S0025-5718-97-00799-0

Preconditioning the Poincaré-Steklov operator by using Green’s function
HTML articles powered by AMS MathViewer

by Jinchao Xu and Sheng Zhang PDF

Math. Comp. 66 (1997), 125-138 Request permission

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

J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp. 47 (1986), no. 175, 103–134. MR 842125, DOI 10.1090/S0025-5718-1986-0842125-3
J. H. Bramble, J. E. Pasciak, and A. H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp. 46 (1986), no. 174, 361–369. MR 829613, DOI 10.1090/S0025-5718-1986-0829613-0
J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. II, Math. Comp. 49 (1987), no. 179, 1–16. MR 890250, DOI 10.1090/S0025-5718-1987-0890250-4
James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz, The construction of preconditioners for elliptic problems by substructuring. IV, Math. Comp. 53 (1989), no. 187, 1–24. MR 970699, DOI 10.1090/S0025-5718-1989-0970699-3
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.
M. Dryja, A method of domain decomposition for three-dimensional finite element elliptic problems, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 43–61. MR 972511
Maksymilian Dryja and Olof B. Widlund, Domain decomposition algorithms with small overlap, SIAM J. Sci. Comput. 15 (1994), no. 3, 604–620. Iterative methods in numerical linear algebra (Copper Mountain Resort, CO, 1992). MR 1273155, DOI 10.1137/0915040
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
J. Carrier, L. Greengard, and V. Rokhlin, A fast adaptive multipole algorithm for particle simulations, SIAM J. Sci. Statist. Comput. 9 (1988), no. 4, 669–686. MR 945931, DOI 10.1137/0909044
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177
A. M. Matsokin and S. V. Nepomnyashchikh. A Schwarz alternating method in a subspace. Soviet Mathematics, 29:78–84, 1985.
—, Norms in the space of traces of mesh functions. Soviet J. Numer. Anal. Math. Modelling, 3:199–216, 1988.
Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584
Barry F. Smith, An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems, SIAM J. Sci. Statist. Comput. 13 (1992), no. 1, 364–378. MR 1145191, DOI 10.1137/0913019
Barry F. Smith and Olof B. Widlund, A domain decomposition algorithm using a hierarchical basis, SIAM J. Sci. Statist. Comput. 11 (1990), no. 6, 1212–1220. MR 1068505, DOI 10.1137/0911069
Charles H. Tong, Tony F. Chan, and C.-C. Jay Kuo, A domain decomposition preconditioner based on a change to a multilevel nodal basis, SIAM J. Sci. Statist. Comput. 12 (1991), no. 6, 1486–1495. MR 1129659, DOI 10.1137/0912082
J. Xu. Theory of multilevel methods. Ph.D. thesis, Cornell University, May 1989.
Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. MR 1193013, DOI 10.1137/1034116
—, Multigrid and domain decomposition methods. Research Monograph (preprint), 1995.