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http://dx.doi.org/10.14317/jami.2015.033

MULTI-PARAMETERIZED SCHWARZ ALTERNATING METHOD FOR 3D-PROBLEM  

Kim, Sang-Bae (Department of Mathematics, Hannam University)
Publication Information
Journal of applied mathematics & informatics / v.33, no.1_2, 2015 , pp. 33-44 More about this Journal
Abstract
The convergence rate of a numerical procedure based on Schwarz Alternating Method(SAM) for solving elliptic boundary value problems depends on the selection of the interface conditions applied on the interior boundaries of the overlapping subdomains. It has been observed that the Robin condition (mixed interface condition), controlled by a parameter, can optimize SAM's convergence rate. In [7], one formulated the multi-parameterized SAM and determined the optimal values of the multi-parameters to produce the best convergence rate for one-dimensional elliptic boundary value problems. Two-dimensional implementation was presented in [8]. In this paper, we present an implementation for three-dimensional problem.
Keywords
elliptic partial differential equations; Schwarz alternating method; Jacobi iterative methods;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 K. Miller, Numerical analogs to the Schwarz alternating procedure, Numer. Math. 7 (1965), 91-103.   DOI
2 J. Oliger, W. Skamarock, and W.P. Tang, Schwarz alternating methods and its SOR accelerations, Tech. report, Department of Computer Science, Stanford University, 1986.
3 G. Rodrigue, Inner/outer iterative methods and numerical Schwarz algorithms, J. Parallel Computing 2 (1985), 205-218.   DOI   ScienceOn
4 G. Rodrigue and P. Saylor, Inner/outer iterative methods and numerical Schwarz algorithms-ii, Proceedings of the IBM Conference on Vector and Parallel Computations for Scientific Computing, IBM, 1985.
5 G. Rodrigue and S. Shah, Pseudo-boundary conditions to accelerate parallel Schwarz methods, Parallel Supercomputing : Methods, Algorithms, and Applications (New York) (G. Carey, ed.), Wiley, 1989, pp. 77-87.
6 G. Rodrigue and J. Simon, A generalization of the numerical Schwarz algorithm, Computing Methods in Applied Sciences and Engineering VI (Amsterdam,New York,Oxford) (R. Glowinski and J. Lions, eds.), North-Holland, 1984, pp. 273-281.
7 G. Rodrigue and J. Simon, Jacobi splitting and the method of overlapping domains for solving elliptic PDE's, Advances in Computer Methods for Partial Differential Equations V (R. Vichnevetsky and R. Stepleman, eds.), IMACS, 1984, pp. 383-386.
8 W.P. Tang, Schwarz Splitting and Template Operators, Ph.D. thesis, Department of Computer Science, Stanford University, 1987.
9 W.P. Tang, Generalized Schwarz splittings, SIAM J. Sci. Stat. Comput. 13 (1992), 573-595.   DOI
10 R. Courant and D. Hilbert, Methods of mathematical physics, vol 2, Willey, New York, 1962.
11 M. Dryja, An additive Schwarz algorithm for two- and three-dimensional finite element elliptic problems, Domain Decomposition Methods (T. Chan, R. Glowinski, J. Periaux, and O. Widlund, eds.), SIAM, 1989, pp. 168-172.
12 D.J. Evans, L.-S. Kang, Y.-P. Chen, and J.-P. Shao, The convergence rate of the Schwarz alternating procedure (iv) : With pseudo-boundary relaxation factor, Intern. J. Computer Math. 21 (1987), 185-203.   DOI   ScienceOn
13 P.R. Halmos, Finite-dimensional Vector Spaces, Van Nostrand, Princeton, N.J., 1958.
14 L.-S. Kang, Domain decomposition methods and parallel algorithms, Second International Symposium on Domain Decomposition Methods for Partial Differential Equations (Philadelphia, PA) (Tony F. Chan, Roland Glowinski, Jacques Periaux, and Olof B. Widlund, eds.), SIAM, 1989, pp. 207-218.
15 L.V. Kantorovich and V.I. Krylov, Approximate methods of higher analysis, 4th ed., P. Noordhoff Ltd, Groningen, The Netherlands, 1958.
16 S.-B. Kim, A. Hadjidimos, E.N. Houstis, and J.R. Rice, Multi-parametrized schwarz splittings for elliptic boundary value problems, Math. and Comp. in Simulation 42 (1996), 47-76.   DOI   ScienceOn
17 S.-B. Kim, Two-dimensional Muti-Parameterized Schwarz Alternating Method, J. Appl. Math. & Informatics 29 (2011), 161-171
18 R.E. Lynch, J.R. Rice, and D.H. Thomas, Direct solution of partial difference equations by tensor products, Numer. Math. 6 (1964), 185-189.   DOI