Abstract
We consider Quadratic Spline Collocation (QSC) methods for linear second order elliptic Partial Differential Equations (PDEs). The standard formulation of these methods leads to non-optimal approximations. In order to derive optimal QSC approximations, high order perturbations of the PDE problem are generated. These perturbations can be applied either to the PDE problem operators or to the right sides, thus leading to two different formulations of optimal QSC methods. The convergence properties of the QSC methods are studied. OptimalO(h 3−j) global error estimates for thejth partial derivative are obtained for a certain class of problems. Moreover,O(h 4−j) error bounds for thejth partial derivative are obtained at certain sets of points. Results from numerical experiments verify the theoretical behaviour of the QSC methods. Performance results also show that the QSC methods are very effective from the computational point of view. They have been implemented efficiently on parallel machines.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlberg, J. H., and T. Ito,A collocation method for two-point boundary value problems, Math. Comp., 29 (1975), pp. 129–131, 761–776.
Archer, D. A.,Some collocation methods for differential equations, Rice University, Houston, TX, U.S.A., Ph.D. thesis, 1973.
Bonomo, J., W. R. Dyksen and J. R. Rice,The ELLPACK performance evaluation system, Purdue University, Computer Science Department, Tech. Rep. CSD-TR 569 (1986), 23 pages.
Cavendish, J. C.,A collocation method for elliptic and parabolic boundary value problems, using cubic splines, University of Pittsburgh, PA, U.S.A., Ph.D. thesis, 1972.
Christara, C. C.,Spline collocation methods, software and architectures for linear elliptic boundary value problems, Ph.D. thesis, Purdue University, IN, U.S.A., 1988.
Christara, C. C. and E. N. Houstis,A domain decomposition spline collocation method for elliptic partial differential equations, Proceedings of the fourth Conference on Hypercubes, Concurrent Computers and Applications (HCCA4), March 1989, Monterey, CA, U.S.A., pp. 1267–1273.
Christara, C. C.,Quadratic spline collocation methods for elliptic PDEs, Univ. of Toronto, Tech. Rep. DCS-135 (1990), (32 pgs).
Christara, C. C.,Conjugate gradient methods for spline collocation equations, Proceedings of the fifth Distributed Memory Computing Conference (DMCC5), April 1990, Charleston, SC, U.S.A., pp. 550–558.
Christara, C. C.,Schur complement preconditioned conjugate gradient methods for spline collocation equations, Proceedings of the 1990 International Conference on Supercomputing (ICS90), June 1990, Amsterdam, the Netherlands, pp. 108–120.
Christara, C. C.,Parallel Solvers for Spline Collocation Equations, Copper Mountain Conference on Iterative Methods, April 1992, CO, U.S.A., submitted for publication, (21 pages).
Daniel, J. W. and B. K. Swartz,Extrapolated collocation for two-point boundary value problems using cubic splines, J. Inst. Maths Applics, 16 (1975), pp. 161–174.
de Boor, C., and B. Swartz,Collocation at Gaussian points, SIAM J. Numer. Anal., 10, 4 (1973), pp. 582–606.
Fyfe, D. J.,The use of cubic splines in the solution of two-point boundary value problems, Comput. J., 17 (1968), pp. 188–192.
Houstis, E. N., E. A. Vavalis and J. R. Rice,Convergence of an O(h 4)cubic spline collocation method for elliptic partial differential equations, SIAM J. Numer. Anal., 25, 1 (1988), pp. 54–74.
Houstis, E. N., C. C. Christara and J. R. Rice,Quadratic spline collocation methods for two-point boundary value problems, Internat. J. Numer. Methods Engrg., 26, (1988), pp. 935–952.
Houstis, E. N., J. R. Rice, C. C. Christara and E. A. Vavalis,Performance of Scientific Software, Mathematical Aspects of Scientific Software, (J. R. Rice, ed.), Springer Verlag, 1988, pp. 123–156.
Irodotou-Ellina, M.,Optimal spline collocation methods for high degree two-point boundary value problems, Aristotle University of Thessaloniki, Greece, Ph.D. thesis, 1987.
Irodotou-Ellina, M. and E. N. Houstis,An O (h 6)quintic spline collocation method for fourth order two-point boundary value problems, BIT, 28 (1988), pp. 288–301.
Kammerer, W. J., G. W. Reddien and R. S. Varga,Quadratic interpolatory splines, Numer. Math., 22 (1974), pp. 241–259.
Khalifa, A. K. and J. C. Eilbeck,Collocation with quadratic and cubic splines, IMA J. Numer. Anal., 2 (1982), pp. 111–121.
Marsden, M. J.,Quadratic spline interpolation, Bull. Amer. Math. Soc., 30 (1974), pp. 903–906.
Papamichael, N., and M. J. Soares,A class of cubic and quintic modified collocation methods for the solution of two-point boundary value problems, Brunel University, Tech. Rep. (1987), 41 pages.
Rice, J. R., E. N. Houstis and W. R. Dyksen,A population of linear second order, elliptic partial differential equations on rectangular domains, Math. Comp., 36 (1981), pp. 475–484.
Rice, J. R. and R. F. Boisvert,Solving Elliptic Problems with ELLPACK, Springer-Verlag, New York, 1985.
Russell, R. D. and L. F. Shampine,A collocation method for boundary value problems, Numer. Math., 19 (1972),pp. 1–28.
Sakai, M. andR. Usmani, Quadratic spline solutions and two-point boundary value problems, Publ. Res. Inst. Math. Sci., Kyoto University, 19 (1983), pp. 7–13.
Author information
Authors and Affiliations
Additional information
This research was supported in part by David Ross Foundation (U.S.A) and NSERC (Natural Sciences and Engineering Research Council of Canada).
Rights and permissions
About this article
Cite this article
Christara, C.C. Quadratic spline collocation methods for elliptic partial differential equations. BIT 34, 33–61 (1994). https://doi.org/10.1007/BF01935015
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01935015