A computational study of finite element methods for second order linear two-point boundary value problems

Keast, P.; Fairweather, G.; Diaz, J. C.

doi:10.1090/S0025-5718-1983-0689467-X

A computational study of finite element methods for second order linear two-point boundary value problems
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by P. Keast, G. Fairweather and J. C. Diaz PDF

Math. Comp. 40 (1983), 499-518 Request permission

Corrigendum: Math. Comp. 43 (1984), 347.

Abstract:

A computational study of five finite element methods for the solution of a single second order linear ordinary differential equation subject to general linear, separated boundary conditions is described. In each method, the approximate solution is a piecewise polynomial expressed in terms of a B-spline basis, and is determined by solving a system of linear algebraic equations with an almost block diagonal structure. The aim of the investigation is twofold: to determine if the theoretical orders of convergence of the methods are realized in practice, and to compare the methods on the basis of cost for a given accuracy. In this study three parametrized families of test problems, containing problems of varying degrees of difficulty, are used. The conclusions drawn are rather straightforward. Collocation is the cheapest method for a given accuracy, and the easiest to implement. Also, for solving the linear algebraic equations, the use of a special purpose solver which takes advantage of the structure of the equations is advisable.

References

U. Ascher, J. Christiansen, and R. D. Russell, A collocation solver for mixed order systems of boundary value problems, Math. Comp. 33 (1979), no. 146, 659–679. MR 521281, DOI 10.1090/S0025-5718-1979-0521281-7
Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR 507062
Carl de Boor and Blâir Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973), 582–606. MR 373328, DOI 10.1137/0710052
Carl de Boor, Efficient computer manipulation of tensor products, ACM Trans. Math. Software 5 (1979), no. 2, 173–182. MR 531913, DOI 10.1145/355826.355831
Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
Julio César Díaz, A collocation-Galerkin method for the two point boundary value problem using continuous piecewise polynomial spaces, SIAM J. Numer. Anal. 14 (1977), no. 5, 844–858. MR 483480, DOI 10.1137/0714057

FORTRAN Packages for Solving Certain Almost Block Diagonal Linear Systems by Modified Alternate Row and Column Elimination

Jim Douglas Jr. and Todd Dupont, Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces, Numer. Math. 22 (1974), 99–109. MR 362922, DOI 10.1007/BF01436724
U. Ascher, S. Pruess, and R. D. Russell, On spline basis selection for solving differential equations, SIAM J. Numer. Anal. 20 (1983), no. 1, 121–142. MR 687372, DOI 10.1137/0720009
Jim Douglas Jr., Todd Dupont, and Lars Wahlbin, Optimal $L_{\infty }$ error estimates for Galerkin approximations to solutions of two-point boundary value problems, Math. Comp. 29 (1975), 475–483. MR 371077, DOI 10.1090/S0025-5718-1975-0371077-0
Jim Douglas Jr., Todd Dupont, and Mary Fanett Wheeler, Some superconvergence results for an $H^{1}$-Galerkin procedure for the heat equation, Computing methods in applied sciences and engineering (Proc. Internat. Sympos., Versailles, 1973) Lecture Notes in Comput. Sci., Vol. 10, Springer, Berlin, 1974, pp. 288–311. MR 0451774
Roderick J. Dunn Jr. and Mary Fanett Wheeler, Some collocation-Galerkin methods for two-point boundary value problems, SIAM J. Numer. Anal. 13 (1976), no. 5, 720–733. MR 433896, DOI 10.1137/0713059
Todd Dupont, A unified theory of superconvergence for Galerkin methods for two-point boundary problems, SIAM J. Numer. Anal. 13 (1976), no. 3, 362–368. MR 408256, DOI 10.1137/0713032
P. W. Hemker, A numerical study of stiff two-point boundary problems, Mathematical Centre Tracts, No. 80, Mathematisch Centrum, Amsterdam, 1977. MR 0488784
Carl de Boor (ed.), Mathematical aspects of finite elements in partial differential equations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Publication No. 33 of the Mathematics Research Center, The University of Wisconsin-Madison. MR 0349031
G. W. Reddien, Projection methods for two-point boundary value problems, SIAM Rev. 22 (1980), no. 2, 156–171. MR 564561, DOI 10.1137/1022025
Robert D. Russell, A comparison of collocation and finite differences for two-point boundary value problems, SIAM J. Numer. Anal. 14 (1977), no. 1, 19–39. MR 451745, DOI 10.1137/0714003
R. D. Russell and J. M. Varah, A comparison of global methods for linear two-point boundary value problems, Math. Comput. 29 (1975), no. 132, 1007–1019. MR 0388788, DOI 10.1090/S0025-5718-1975-0388788-3
J. Stoer and R. Bulirsch, Introduction to numerical analysis, Springer-Verlag, New York-Heidelberg, 1980. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. MR 557543
James M. Varah, A comparison of some numerical methods for two-point boundary value problems, Math. Comp. 28 (1974), 743–755. MR 373300, DOI 10.1090/S0025-5718-1974-0373300-4
J. M. Varah, Alternate row and column elimination for solving certain linear systems, SIAM J. Numer. Anal. 13 (1976), no. 1, 71–75. MR 411199, DOI 10.1137/0713008
Mary Fanett Wheeler, An optimal $L_{\infty }$ error estimate for Galerkin approximations to solutions of two-point boundary value problems, SIAM J. Numer. Anal. 10 (1973), 914–917. MR 343659, DOI 10.1137/0710077
Mary Fanett Wheeler, A $C^{0}$-collocation-finite element method for two-point boundary value problems and one space dimensional parabolic problems, SIAM J. Numer. Anal. 14 (1977), no. 1, 71–90. MR 455429, DOI 10.1137/0714005

Evaluation of B-Splines for Solving Systems of Boundary Value Problems

A Comparative Study of Finite Element Methods for the Solution of Second Order Linear Two-Point Boundary Value Problems

R. Russell, Efficiencies of $B$-spline methods for solving differential equations, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) Congressus Numerantium, No. XVI, Utilitas Math. Publ., Winnipeg, Man., 1976, pp. 599–617. MR 0405871

ROWCOL-A Package for Solving Almost Block Diagonal Linear Systems Arising in

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