## Analysis of non-overlapping domain decomposition algorithms with inexact solves

HTML articles powered by AMS MathViewer

- by James H. Bramble, Joseph E. Pasciak and Apostol T. Vassilev PDF
- Math. Comp.
**67**(1998), 1-19 Request permission

## Abstract:

In this paper we construct and analyze new non-overlapping domain decomposition preconditioners for the solution of second-order elliptic and parabolic boundary value problems. The preconditioners are developed using uniform preconditioners on the subdomains instead of exact solves. They exhibit the same asymptotic condition number growth as the corresponding preconditioners with exact subdomain solves and are much more efficient computationally. Moreover, this asymptotic condition number growth is bounded independently of jumps in the operator coefficients across subdomain boundaries. We also show that our preconditioners fit into the additive Schwarz framework with appropriately chosen subspace decompositions. Condition numbers associated with the new algorithms are computed numerically in several cases and compared with those of the corresponding algorithms in which exact subdomain solves are used.## References

- 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**, DOI 10.1137/0723075 - Christoph Börgers,
*The Neumann-Dirichlet domain decomposition method with inexact solvers on the subdomains*, Numer. Math.**55**(1989), no. 2, 123–136. MR**987381**, DOI 10.1007/BF01406510 - J. H. Bramble, C. I. Goldstein, and J. E. Pasciak,
*Analysis of $V$-cycle multigrid algorithms for forms defined by numerical quadrature*, SIAM J. Sci. Comput.**15**(1994), no. 3, 566–576. Iterative methods in numerical linear algebra (Copper Mountain Resort, CO, 1992). MR**1273152**, DOI 10.1137/0915037 - 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. 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,
*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. III*, Math. Comp.**51**(1988), no. 184, 415–430. MR**935071**, DOI 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 10.1090/S0025-5718-1989-0970699-3 - James H. Bramble, Joseph E. Pasciak, Jun Ping Wang, and Jinchao Xu,
*Convergence estimates for product iterative methods with applications to domain decomposition*, Math. Comp.**57**(1991), no. 195, 1–21. MR**1090464**, DOI 10.1090/S0025-5718-1991-1090464-8 - James H. Bramble, Joseph E. Pasciak, and Jinchao Xu,
*Parallel multilevel preconditioners*, Math. Comp.**55**(1990), no. 191, 1–22. MR**1023042**, DOI 10.1090/S0025-5718-1990-1023042-6 - J.H. Bramble, J.E. Pasciak and J. Xu,
*A multilevel preconditioner for domain decomposition boundary systems*, Proceedings of the 10’th Inter. Conf. on Comput. Meth. in Appl. Sci. and Engr., Nova Sciences, New York, 1992. - James H. Bramble and Jinchao Xu,
*Some estimates for a weighted $L^2$ projection*, Math. Comp.**56**(1991), no. 194, 463–476. MR**1066830**, DOI 10.1090/S0025-5718-1991-1066830-3 - Lawrence C. Cowsar, Jan Mandel, and Mary F. Wheeler,
*Balancing domain decomposition for mixed finite elements*, Math. Comp.**64**(1995), no. 211, 989–1015. MR**1297465**, DOI 10.1090/S0025-5718-1995-1297465-9 - M. Dryja,
*A capacitance matrix method for Dirichlet problem on polygon region*, Numer. Math.**39**(1982), no. 1, 51–64. MR**664536**, DOI 10.1007/BF01399311 - 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, Barry F. Smith, and Olof B. Widlund,
*Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions*, SIAM J. Numer. Anal.**31**(1994), no. 6, 1662–1694. MR**1302680**, DOI 10.1137/0731086 - 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 - Ruth Gonzalez and Mary Fanett Wheeler,
*Domain decomposition for elliptic partial differential equations with Neumann boundary conditions*, Proceedings of the international conference on vector and parallel computing—issues in applied research and development (Loen, 1986), 1987, pp. 257–263. MR**898046**, DOI 10.1016/0167-8191(87)90022-6 - G. Haase, U. Langer, and A. Meyer,
*The approximate Dirichlet domain decomposition method. I. An algebraic approach*, Computing**47**(1991), no. 2, 137–151 (English, with German summary). MR**1139433**, DOI 10.1007/BF02253431 - S.V. Nepomnyaschikh,
*Application of domain decomposition to elliptic problems with discontinuous coefficients*, Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, (eds, R. Glowinski, Y.A. Kuznetzov, G.A. Meurant, and J. Périaux) SIAM, Phil. PN, 1991, pp. 242–251. - B.F. Smith,
*Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity*, Ph.D. Thesis, Courant Institute of Mathematical Sciences, Dept. of Computer Science Tech. Rep. 517, New York, 1990. - A.T. Vassilev,
*On Discretization and Iterative Techniques for Second–Order Problems with Applications to Multiphase Flow in Porous Media*, Ph.D. Thesis, Texas A&M University, College Station, Texas, 1996. - Olof B. Widlund,
*Iterative substructuring methods: algorithms and theory for elliptic problems in the plane*, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 113–128. MR**972514** - Jinchao Xu,
*Iterative methods by space decomposition and subspace correction*, SIAM Rev.**34**(1992), no. 4, 581–613. MR**1193013**, DOI 10.1137/1034116

## Additional Information

**James H. Bramble**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- Email: bramble@math.tamu.edu
**Joseph E. Pasciak**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- Email: pasciak@math.tamu.edu
**Apostol T. Vassilev**- Affiliation: Schlumberger, 8311 N. FM 620 Rd., Austin, Texas 78726
- Email: vassilev@slb.com
- Received by editor(s): February 21, 1996
- Received by editor(s) in revised form: September 6, 1996
- Additional Notes: This manuscript has been authored under contract number DE-AC02-76CH00016 with the U.S. Department of Energy. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. This work was also supported in part under the National Science Foundation Grant No. DMS-9007185 and by the PICS ground water research initiative under contract AS-413-ASD.
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp.
**67**(1998), 1-19 - MSC (1991): Primary 65N30, 65F10
- DOI: https://doi.org/10.1090/S0025-5718-98-00879-5
- MathSciNet review: 1432125