Convergence estimates for product iterative methods with applications to domain decomposition

Authors:
James H. Bramble, Joseph E. Pasciak, Jun Ping Wang and Jinchao Xu

Journal:
Math. Comp. **57** (1991), 1-21

MSC:
Primary 65J10; Secondary 65M55, 65N22, 65N55

DOI:
https://doi.org/10.1090/S0025-5718-1991-1090464-8

MathSciNet review:
1090464

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider iterative methods for the solution of symmetric positive definite problems on a space $\mathcal {V}$ which are defined in terms of products of operators defined with respect to a number of subspaces. The simplest algorithm of this sort has an error-reducing operator which is the product of orthogonal projections onto the complement of the subspaces. New norm-reduction estimates for these iterative techniques will be presented in an abstract setting. Applications are given for overlapping Schwarz algorithms with many subregions for finite element approximation of second-order elliptic problems.

- Jean-Pierre Aubin,
*Approximation of elliptic boundary-value problems*, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1972. Pure and Applied Mathematics, Vol. XXVI. MR**0478662** - 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**
I. Babuška, - Garrett Birkhoff and Arthur Schoenstadt (eds.),
*Elliptic problem solvers. II*, Academic Press, Inc., Orlando, FL, 1984. MR**764219** - James H. Bramble and Joseph E. Pasciak,
*New convergence estimates for multigrid algorithms*, Math. Comp.**49**(1987), no. 180, 311–329. MR**906174**, DOI https://doi.org/10.1090/S0025-5718-1987-0906174-X - 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 https://doi.org/10.1090/S0025-5718-1986-0842125-3 - 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 https://doi.org/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. III*, Math. Comp.**51**(1988), no. 184, 415–430. MR**935071**, DOI https://doi.org/10.1090/S0025-5718-1988-0935071-X - 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 https://doi.org/10.1090/S0025-5718-1989-0970699-3
J. H. Bramble, J. E. Pasciak, J. Wang, and J. Xu, - James H. Bramble, Joseph E. Pasciak, and Jinchao Xu,
*Parallel multilevel preconditioners*, Math. Comp.**55**(1990), no. 191, 1–22. MR**1023042**, DOI https://doi.org/10.1090/S0025-5718-1990-1023042-6 - Philippe G. Ciarlet,
*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174**
M. Dryja and O. Widlund, - P.-L. Lions,
*On the Schwarz alternating method. I*, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 1–42. MR**972510** - Jan Mandel and Steve McCormick,
*Iterative solution of elliptic equations with refinement: the two-level case*, Domain decomposition methods (Los Angeles, CA, 1988) SIAM, Philadelphia, PA, 1989, pp. 81–92. MR**992005**
T. P. Mathew, - Olof Widlund,
*Some domain decomposition and iterative refinement algorithms for elliptic finite element problems*, J. Comput. Math.**7**(1989), no. 2, 200–208. China-US Seminar on Boundary Integral and Boundary Element Methods in Physics and Engineering (Xi’an, 1987–88). MR**1016840**
J. Xu,

*On the Schwarz algorithm in the theory of differential equations of mathematical physics*, Czechoslovak Math. J.

**8**(1958), 328-342 (Russian).

*Multigrid results which do not depend upon elliptic regularity assumptions*(in preparation).

*An additive variant of the Schwarz alternating method for the case of many subregions*, Technical Report 339, Courant Institute of Mathematical Sciences, 1987. ---,

*Some domain decomposition algorithms for elliptic problems*, Technical Report 438, Courant Institute of Mathematical Sciences, 1989.

*Domain decomposition and iterative refinement methods for mixed finite element discretizations of elliptic problems*, Thesis, New York University, 1989. H. A. Schwarz,

*Ueber einige Abbildungsaufgaben*, J. Reine Angew. Math.

**70**(1869), 105-120. [Ges. Math. Abh., vol.2, 65-83].

*Theory of multilevel methods*, Dept. Math. Rep. AM-48, Penn. State University, 1989.

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Additional Information

Keywords:
Second-order elliptic equation,
domain decomposition

Article copyright:
© Copyright 1991
American Mathematical Society