# Quadratic spline collocation methods for elliptic partial differential equations

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## 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. Optimal*O*(*h*^{3−j}) global error estimates for the*j*th partial derivative are obtained for a certain class of problems. Moreover,*O*(*h*^{4−j}) error bounds for the*j*th 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.

## AMS(MOS) subject classifications

65N35 65N15## Key words

spline collocation elliptic partial differential equations second order boundary value problems## Preview

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## References

- 1.Ahlberg, J. H., and T. Ito,
*A collocation method for two-point boundary value problems*, Math. Comp., 29 (1975), pp. 129–131, 761–776.Google Scholar - 2.Archer, D. A.,
*Some collocation methods for differential equations*, Rice University, Houston, TX, U.S.A., Ph.D. thesis, 1973.Google Scholar - 3.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.Google Scholar - 4.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.Google Scholar - 5.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.Google Scholar - 6.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.Google Scholar - 7.Christara, C. C.,
*Quadratic spline collocation methods for elliptic PDEs*, Univ. of Toronto, Tech. Rep. DCS-135 (1990), (32 pgs).Google Scholar - 8.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.Google Scholar - 9.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.Google Scholar - 10.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).Google Scholar - 11.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.Google Scholar - 12.de Boor, C., and B. Swartz,
*Collocation at Gaussian points*, SIAM J. Numer. Anal., 10, 4 (1973), pp. 582–606.Google Scholar - 13.Fyfe, D. J.,
*The use of cubic splines in the solution of two-point boundary value problems*, Comput. J., 17 (1968), pp. 188–192.Google Scholar - 14.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.Google Scholar - 15.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.Google Scholar - 16.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.Google Scholar - 17.Irodotou-Ellina, M.,
*Optimal spline collocation methods for high degree two-point boundary value problems*, Aristotle University of Thessaloniki, Greece, Ph.D. thesis, 1987.Google Scholar - 18.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.Google Scholar - 19.Kammerer, W. J., G. W. Reddien and R. S. Varga,
*Quadratic interpolatory splines*, Numer. Math., 22 (1974), pp. 241–259.Google Scholar - 20.Khalifa, A. K. and J. C. Eilbeck,
*Collocation with quadratic and cubic splines*, IMA J. Numer. Anal., 2 (1982), pp. 111–121.Google Scholar - 21.Marsden, M. J.,
*Quadratic spline interpolation*, Bull. Amer. Math. Soc., 30 (1974), pp. 903–906.Google Scholar - 22.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.Google Scholar - 23.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.Google Scholar - 24.Rice, J. R. and R. F. Boisvert,
*Solving Elliptic Problems with ELLPACK*, Springer-Verlag, New York, 1985.Google Scholar - 25.Russell, R. D. and L. F. Shampine,
*A collocation method for boundary value problems, Numer. Math.*, 19 (1972),*pp*. 1–28.Google Scholar - 26.
*Sakai, M.*and*R. Usmani, Quadratic spline solutions and two-point boundary value problems*, Publ. Res. Inst. Math. Sci., Kyoto University, 19 (1983), pp. 7–13.Google Scholar