Parallel solvers for spline collocation equations

Advances in Engineering Software

Volumn 27, Issue 1-2, 1996, Pages 71-89

  Christara, Christina C ^a  

^a UNIVERSITY OF TORONTO   (Canada)

Author keywords

Distributed memory machines; Domain decomposition; Elliptic partial differential equations; Iterative methods

Indexed keywords

APPROXIMATION THEORY; BOUNDARY VALUE PROBLEMS; COMPUTATION THEORY; CONVERGENCE OF NUMERICAL METHODS; DISTRIBUTED COMPUTER SYSTEMS; FINITE ELEMENT METHOD; ITERATIVE METHODS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS; THREE DIMENSIONAL;

DISTRIBUTED MEMORY MACHINES; ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS; HERMITE SPLINE COLLOCATION METHOD; SCHUR COMPLEMENT METHOD; SPLINE COLLOCATION EQUATIONS;

PARALLEL PROCESSING SYSTEMS;

EID: 0030269961     PISSN: 09659978     EISSN: None     Source Type: Journal    
DOI: 10.1016/0965-9978(96)00019-1     Document Type: Article

References (32)

1. Ahlberg, J. H. & Ito, T., A collocation method for two-point boundary value problems. Math. Comput., 1975, 29, 129-131, 761-776.
2. Fyfe, D. J., The use of cubic splines in the solution of two-point boundary value problems. Comput. J., 1968, 17, 188-192.
3. Archer, D. A., Some collocation methods for differential equations. PhD thesis, Rice University, Houston, TX, 1973.
4. Daniel, J. W. & Swartz, B. K., Extrapolated collocation for two-point boundary value problems using cubic splines. J. Inst. Maths Appl., 1975, 16, 161-174.
5. Papamichael, N. & Soares, M. J., A class of cubic and quintic spline modified collocation methods for the solution of two-point boundary value problems. Tech. report, Brunel University, 1987.
6. Irodotou-Ellina, M., Optimal spline collocation methods for high degree two-point boundary value problems. PhD thesis, University of Thessaloniki, Greece, 1987.
7. 6) quintic spline collocation method for fourth order two-point boundary value problems. BIT, 1988, 28, 288-301.
8. Christara, C. C., Spline collocation methods, software and architectures for linear elliptic boundary value problems. PhD thesis, Purdue University, IN, 1988.
9. 4) cubic spline collocation method for elliptic partial differential equations. SIAM J. Numer. Anal., 1988, 25(1), 54-74.
10. Houstis, E. N., Christara, C. C. & Rice, J. R., Quadratic spline collocation methods for two-point boundary value problems. Int. J. Numer. Meth. Engng, 1988, 26, 935-952.
11. Houstis, E. N., Rice, J. R., Christara, C. C. & Vavalis, E. A., Performance of scientific software. In Mathematical Aspects of Scientific Software, ed. J. R. Rice. Springer, New York, 1988, pp. 123-156.
12. Christara, C. C., Quadratic spline collocation methods for elliptic partial differential equations. BIT, 1994, 34(1), 33-61.
13. Lions, P.-L., On the Schwarz alternating method III: a variant for nonoverlapping subdomains. In Domain Decomposition Methods for Partial Differential Equations, eds T. Chan, R. Glowinski, J. Periaux & O. Widlund. SIAM, 1990, pp. 202-223.
14. Ehrlich, L. W., The numerical Schwarz alternating procedure and SOR. SIAM J. Sci. Statist. Comput., 1986, 7(3), 989-993.
15. Rodrigue, G. & Simon, J., A generalisation of the numerical Schwarz algorithm. In Computing Methods in Applied Science and Engineering VI, eds R. Glowinski & J.-L. Lions. Elsevier Science, Amsterdam, 1984, pp. 273-283.
16. Dihn, Q. V., Glowinski, R. & Periaux, J., Solving elliptic problems by domain decomposition methods with applications. In Elliptic Problem Solvers II. Academic Press, New York, 1989, pp. 395-426.
17. Dryja, M., An additive Schwarz algorithm for two-and three-dimensional finite element elliptic problems. In Domain Decomposition Methods, eds T. Chan, R. Glowinski, J. Periaux & O. Widlund. SIAM, 1989, pp. 168-172.
18. Christara, C. C. & Smith, B., Convergence of quadratic spline collocation on the union of rectangular subdomains (in preparation).
19. Gropp, W. D. & Smith, B., Users manual for KSP: data-structure-neutral codes implementing Krylov space methods. Tech. report ANL-93/30, Argonne National Laboratory, 1993.
20. Bjorstad, P. E. & Widlund, O. B., Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal., 1986, 23(6), 1097-1120.
21. Bramble, J. H., Pasciak, J. E. & Schatz, A. H., The construction of preconditioners for elliptic problems by substructuring I. Math. Comput., 1986, 47, 103-134.
22. Dryja, M., A finite element-capacitance method for elliptic problems on regions partitioned into subregions. Numer. Math., 1984, 44, 153-168.
23. Dryja, M. & Proskurowski, W., Iterative methods in subspaces for solving elliptic problems using domain decomposition. Tech. report CRI-86-10, University of Southern California, 1986.
24. Keyes, D. E. & Gropp, W. D., A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation. SIAM J. Sci. Statist. Comput., 1987, 8, s166-s202.
25. Mandel, J., On block diagonal and Schur complement preconditioning. Numer. Math., 1990, 58, 79-93.
26. Christara, C. C., Schur complement preconditioned conjugate gradient methods for spline collocation equations. In Proc. 1990 Int. Conf. Supercomputing (ICS90), Amsterdam, the Netherlands, June 1990, pp. 108-120.
27. Kightley, J. R. & Forsyth, P. A. (rev. E. F. D'Azevedo), MAT1 iterative sparse matrix solver: user's guide. University of Waterloo, 1989.
28. Oppe, T. C., Joubert, W. D. & Kincaid, D. R., NSPCG User's Guide, Version 1.0. Center for Numerical Analysis, the University of Texas at Austin, CNA-216, April 1988.
29. Christara, C. C. & Smith, B., Multilevel methods for quadratic spline collocation. BIT (submitted).
30. Christara, C. C., Conjugate gradient methods for spline collocation equations. In Proc. 5th Distributed Memory Computing Conf. (DMCC5), Charleston, SC, April 1990, pp. 550-558.
31. Christara, C. C., Parallel computation of partial differential equations on distributed memory machines. In Advances in Computer Methods for Partial Differential Equations, VII, eds R. Vichnevetsky, D. Knight & G. Richter. (Proc. 7th IMACS Int. Conf. Computer Methods for Partial Differential Equations), Rutgers University, New Brunswick, 22-24 June 1992, pp. 142-149.
32. Duff, I. S. & Meurant, G. A., The effect of ordering on preconditioned conjugate gradients. BIT, 1989, 29, 635-657.