The construction of preconditioners for elliptic problems by substructuring. III

Authors:
James H. Bramble, Joseph E. Pasciak and Alfred H. Schatz

Journal:
Math. Comp. **51** (1988), 415-430

MSC:
Primary 65N30; Secondary 65F10

DOI:
https://doi.org/10.1090/S0025-5718-1988-0935071-X

MathSciNet review:
935071

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Abstract: In earlier parts of this series of papers, we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems. The resulting algorithms are well suited for implementation on computers with parallel architecture. In this paper, we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners. The iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns. These preconditioners involve an incomplete Chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work. Theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods.

**[1]**Petter E. Bjørstad and Olof B. Widlund,*Solving elliptic problems on regions partitioned into substructures*, Elliptic problem solvers, II (Monterey, Calif., 1983) Academic Press, Orlando, FL, 1984, pp. 245–255. MR**764237****[2]**Petter E. Bjørstad and Olof B. Widlund,*Iterative methods for the solution of elliptic problems on regions partitioned into substructures*, SIAM J. Numer. Anal.**23**(1986), no. 6, 1097–1120. MR**865945**, https://doi.org/10.1137/0723075**[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**, https://doi.org/10.1090/S0025-5718-1986-0829613-0**[4]**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**, https://doi.org/10.1090/S0025-5718-1986-0842125-3**[5]**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**, https://doi.org/10.1090/S0025-5718-1987-0890250-4**[6]**B. L. Buzbee and Fred W. Dorr,*The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular regions*, SIAM J. Numer. Anal.**11**(1974), 753–763. MR**0362944**, https://doi.org/10.1137/0711061**[7]**B. L. Buzbee, F. W. Dorr, J. A. George, and G. H. Golub,*The direct solution of the discrete Poisson equation on irregular regions*, SIAM J. Numer. Anal.**8**(1971), 722–736. MR**0292316**, https://doi.org/10.1137/0708066**[8]**Q. V. Dihn, R. Glowinski & J. Périaux, "Solving elliptic problems by domain decomposition methods," in*Elliptic Problem Solvers II*(G. Birkhoff and A. Schoenstadt, eds.), Academic Press, New York, 1984, pp. 395-426.**[9]**G. H. Golub & D. Meyers,*The Use of Preconditioning Over Irregular Regions*, Proc. 6th Internat. Conf. Comput. Meth. Sci. and Engrg., Versailles, 1983.**[10]**J. L. Lions & E. Magenes,*Problèmes aux Limites non Homogènes et Applications*, Dunod, Paris, 1968.**[11]**J. Nečas,*Les Méthodes Directes en Théorie des Équations Elliptiques*, Academia, Prague, 1967.**[12]**W. M. Patterson, 3rd,*Iterative Methods for the Solution of a Linear Operator Equation in Hilbert Space--A Survey*, Lecture Notes in Math., vol. 394, Springer-Verlag, New York, 1974.

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DOI:
https://doi.org/10.1090/S0025-5718-1988-0935071-X

Article copyright:
© Copyright 1988
American Mathematical Society