An iterative method for elliptic problems on regions partitioned into substructures

Authors:
J. H. Bramble, J. E. Pasciak and A. H. Schatz

Journal:
Math. Comp. **46** (1986), 361-369

MSC:
Primary 65N20; Secondary 65F10, 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1986-0829613-0

MathSciNet review:
829613

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Abstract: Some new preconditioners for discretizations of elliptic boundary problems are studied. With these preconditioners, the domain under consideration is broken into subdomains and preconditioners are defined which only require the solution of matrix problems on the subdomains. Analytic estimates are given which guarantee that under appropriate hypotheses, the preconditioned iterative procedure converges to the solution of the discrete equations with a rate per iteration that is independent of the number of unknowns. Numerical examples are presented which illustrate the theoretically predicted iterative convergence rates.

**[1]**O. Axelsson,*A class of iterative methods for finite element equations*, Comput. Methods Appl. Mech. Engrg.**9**(1976), no. 2, 123–127. MR**0433836**, https://doi.org/10.1016/0045-7825(76)90056-6**[2]**Ivo Babuška and A. K. Aziz,*Survey lectures on the mathematical foundations of the finite element method*, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR**0421106****[3]**Garrett Birkhoff and Arthur Schoenstadt (eds.),*Elliptic problem solvers. II*, Academic Press, Inc., Orlando, FL, 1984. MR**764219****[4]**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**[5]**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**[6]**R. Chandra,*Conjugate Gradient Methods for Partial Differential Equations*, Yale Univ. Dept. Comp. Sci. Report No. 129, 1978.**[7]**M. Dryja,*A capacitance matrix method for Dirichlet problem on polygon region*, Numer. Math.**39**(1982), no. 1, 51–64. MR**664536**, https://doi.org/10.1007/BF01399311**[8]**P. Grisvard,*Elliptic problems in nonsmooth domains*, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR**775683****[9]**Paul Concus, Gene H. Golub, and Dianne P. O’Leary,*A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations*, Sparse matrix computations (Proc. Sympos., Argonne Nat. Lab., Lemont, Ill., 1975) Academic Press, New York, 1976, pp. 309–332. MR**0501821****[10]**H. C. Elman,*Iterative Methods for Large, Sparse, Nonsymmetric Systems of Linear Equations*, Yale Univ. Dept. Comp. Sci. Report No. 229, 1978.**[11]**J.-L. Lions and E. Magenes,*Problèmes aux limites non homogènes et applications. Vol. 1*, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR**0247243****[12]**J. Nečas,*Les Méthodes Directes en Théorie des Équations Elliptiques*, Academia, Prague, 1967.**[13]**Włodzimierz Proskurowski and Olof Widlund,*A finite element-capacitance matrix method for the Neumann problem for Laplace’s equation*, SIAM J. Sci. Statist. Comput.**1**(1980), no. 4, 410–425. MR**610753**, https://doi.org/10.1137/0901029**[14]**Wlodzimierz Proskurowski and Olof Widlund,*On the numerical solution of Helmholtz’s equation by the capacitance matrix method*, Math. Comp.**30**(1976), no. 135, 433–468. MR**0421102**, https://doi.org/10.1090/S0025-5718-1976-0421102-4

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DOI:
https://doi.org/10.1090/S0025-5718-1986-0829613-0

Article copyright:
© Copyright 1986
American Mathematical Society