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

Authors:
P. Keast, G. Fairweather and J. C. Diaz

Journal:
Math. Comp. **40** (1983), 499-518

MSC:
Primary 65L10; Secondary 65N30, 65N35

DOI:
https://doi.org/10.1090/S0025-5718-1983-0689467-X

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

MathSciNet review:
689467

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**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**, https://doi.org/10.1090/S0025-5718-1979-0521281-7**[2]**Carl de Boor,*A practical guide to splines*, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR**507062****[3]**Carl de Boor and Blâir Swartz,*Collocation at Gaussian points*, SIAM J. Numer. Anal.**10**(1973), 582–606. MR**373328**, https://doi.org/10.1137/0710052**[4]**V. Pereyra & R. D. Russell "Difficulties of comparing complex mathematical software: general comments and the BVODE case." (Preprint.)**[5]**Carl de Boor,*Efficient computer manipulation of tensor products*, ACM Trans. Math. Software**5**(1979), no. 2, 173–182. MR**531913**, https://doi.org/10.1145/355826.355831**[6]**Earl A. Coddington and Norman Levinson,*Theory of ordinary differential equations*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR**0069338****[7]**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**, https://doi.org/10.1137/0714057**[8]**J. C. Diaz. G. Fairweather & P. Keast,*FORTRAN Packages for Solving Certain Almost Block Diagonal Linear Systems by Modified Alternate Row and Column Elimination*, Technical Report 148/81, Department of Computer Science, University of Toronto, January, 1981.**[9]**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**, https://doi.org/10.1007/BF01436724**[10]**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**, https://doi.org/10.1137/0720009**[11]**Jim Douglas Jr., Todd Dupont, and Lars Wahlbin,*Optimal 𝐿_{∞} error estimates for Galerkin approximations to solutions of two-point boundary value problems*, Math. Comp.**29**(1975), 475–483. MR**371077**, https://doi.org/10.1090/S0025-5718-1975-0371077-0**[12]**Jim Douglas Jr., Todd Dupont, and Mary Fanett Wheeler,*Some superconvergence results for an 𝐻¹-Galerkin procedure for the heat equation*, Computing methods in applied sciences and engineering (Proc. Internat. Sympos., Versailles, 1973) Springer, Berlin, 1974, pp. 288–311. Lecture Notes in Comput. Sci., Vol. 10. MR**0451774****[13]**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**, https://doi.org/10.1137/0713059**[14]**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**, https://doi.org/10.1137/0713032**[15]**G. Fairweather. P. Keast & J. C. Diaz, "On the -Galerkin method for two-point boundary value problems and parabolic problems in one space variable." (In preparation.)**[16]**P. W. Hemker,*A numerical study of stiff two-point boundary problems*, Mathematisch Centrum, Amsterdam, 1977. Mathematical Centre Tracts, No. 80. MR**0488784****[17]**Carl de Boor (ed.),*Mathematical aspects of finite elements in partial differential equations*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Publication No. 33 of the Mathematics Research Center, The University of Wisconsin-Madison. MR**0349031****[18]**G. W. Reddien,*Projection methods for two-point boundary value problems*, SIAM Rev.**22**(1980), no. 2, 156–171. MR**564561**, https://doi.org/10.1137/1022025**[19]**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**, https://doi.org/10.1137/0714003**[20]**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**, https://doi.org/10.1090/S0025-5718-1975-0388788-3**[21]**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****[22]**James M. Varah,*A comparison of some numerical methods for two-point boundary value problems*, Math. Comp.**28**(1974), 743–755. MR**373300**, https://doi.org/10.1090/S0025-5718-1974-0373300-4**[23]**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**, https://doi.org/10.1137/0713008**[24]**Mary Fanett Wheeler,*An optimal 𝐿_{∞} error estimate for Galerkin approximations to solutions of two-point boundary value problems*, SIAM J. Numer. Anal.**10**(1973), 914–917. MR**343659**, https://doi.org/10.1137/0710077**[25]**Mary Fanett Wheeler,*A 𝐶⁰-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**, https://doi.org/10.1137/0714005**[26]**U. Ascher & R. D. Russell,*Evaluation of B-Splines for Solving Systems of Boundary Value Problems*, Computer Sciences Technical Report 77-14, University of British Columbia, Vancouver, 1977.**[27]**P. Keast, G. Fairweather & J. C. Diaz,*A Comparative Study of Finite Element Methods for the Solution of Second Order Linear Two-Point Boundary Value Problems*, Technical Report 150/81, Department of Computer Science, University of Toronto, March 1981.**[28]**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) Utilitas Math. Publ., Winnipeg, Man., 1976, pp. 599–617. Congressus Numerantium, No. XVI. MR**0405871****[29]**G. Fairweather & P. Keast,*ROWCOL-A Package for Solving Almost Block Diagonal Linear Systems Arising in*-*Galerkin and Collocation*- -*Galerkin Methods*, Technical Report 158/82. Department of Computer Science, University of Toronto, 1982.

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DOI:
https://doi.org/10.1090/S0025-5718-1983-0689467-X

Article copyright:
© Copyright 1983
American Mathematical Society