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The finite difference versus the finite element method for the solution of boundary value problems

Published online by Cambridge University Press:  17 April 2009

Vidar Thomée
Vidar Thomée
Affiliation:
Department of Mathematics, Chalmers University of Technology, Göteborg, Sweden.
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In this lecture we describe, discuss and compare the two classes of methods most commonly used for the numerical solution of boundary value problems for partial differential equations, namely, the finite difference method and the finite element method. For both of these methods an extensive development of mathematical error analysis has taken place but individual numerical analysts often express strong prejudices in favor of one of them. Our purpose is to try to convey our conviction that this attitude is both historically unjustified and inhibiting, and that familiarity with both methods provides a wider range of techniques for constructing and analyzing discretization schemes.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 29 , Issue 2 , April 1984 , pp. 267 - 288
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Argyris, J.H., Energy theorems and structural analysis. A generalised discourse with applications on energy proinciples of structural analysis including the effects of temperature and non-linear stress-strain relations (Butterworths, London, Toronto, Sydney, 1960).Google Scholar
[2]Aubin, Jean Pierre, “Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin's and finite difference methods”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 21 (1967), 599–637.Google Scholar
[3]Birkhoff, G., Schultz, M.H., and Varga, R.S., “Piecewise Hermite interpolation in one and two variables with applications to partial differential equations”, Numer. Math. 11 (1968), 232–256.CrossRefGoogle Scholar
[4]Céa, Jean, “Approximation variationelle des problèmes aux limites”, Ann. Inst. Fourier (Grenoble) 14 (1964), 345–444.CrossRefGoogle Scholar
[5]Chen, Chuan-miao and Thomée, Vidar, “The lumped mass finite element method for a parabolic problem” (Report No. 1983–1. Department of Mathematics, GoteborgGöteborg, Sweden).CrossRefGoogle Scholar
[6]Ciarlet, Philippe G., The finite element method for elliptic problems (Studies in Mathematics and its Applications, 4. North-Holland, Amsterdam, New York, Oxford, 1978).Google Scholar
[7]Clough, R.W., “The finite element method in plane stress analysis”,Proc. Second ASCE Conf. on Electronic Computation,1960,345–378 (Pittsberg, pennsylvania, 1960).Google Scholar
[8]Courant, R., “Variational methods for the solution of problems of equilibrium and vibrations”, Bull. Amer. Math. Soc. 49 (1943), 1–23.CrossRefGoogle Scholar
[9]Courant, R., Friedrichs, K. und Lewy, H., “Über die partiellen Differenzengleichungen der Mathematischen Physik”, Math. Ann. 100 (1928), 32–74.CrossRefGoogle Scholar
[10]Delfour, M., Hager, W. and Trochu, F., “Discontinuous Galerkin methods for ordinary differential equations”, Math. Comp. 36 (1981), 455–473.CrossRefGoogle Scholar
[11] Ю.К. Демьянович [Demjanovič, Ju.K.], “метод сеток для некоторых задач математической Φазикн” [The net method for some problems in mathematical physics], Dokl. Akad. Nauk SSSR 159 (1964), 250–253.Google Scholar
[12]Douglas, Jim Jr. and Dupont, Todd, “Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary value problems”, Topics in numerical analysis, 89–92 (Proc. Roy. Irish Acad. Conf.,University College,Dublin,1972. Academic Press, New York, London, 1973).Google Scholar
[13]Eriksson, K., Johnson, C. and Thomée, Vidar, “The discontinuous Galerkin method for time discretization of parabolic problems” (in preparation).Google Scholar
[14]Eriksson, K. and Thomée, Vidar, “Galerkin methods for singular boundary value problems in one space dimension” (Repart No. 1982–11. Department of Mathematics, Goteborg, Sweden, 1982).Google Scholar
[15]Faedo, Sandro, “Un nuovo metodo der l'analisi esistenziale e quantitativa dei problemi di propagazione”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 1 (1949), 1–40.Google Scholar
[16]Feng, K., “Finite difference schemes based on variational principles” (Chinese), Appl. Math. Comput. Math. 2 (1965), 238–262.Google Scholar
[17]Friedrichs, K.O. and Keller, H.B., “A finite difference scheme for generalized Neumann problems”, Numerical solution of partial differential equations, 1–19 (Academic Press, New York, London, 1966).Google Scholar
[18]Fujii, H., “Some remarks on finite element analysis of time-dependent field problems”, Theory and practice in finite element structural analysis, 91–106 (University of Tokyo Press, Tokyo, 1973).Google Scholar
[19]Galerkin, B.G., “Rods and plates. Series occurring in various questions concerning the elastic equilibrium of rods and plates”, Vestnik Inzh. 19 (1915), 897–908.Google Scholar
[20]Gerschgorin, S., “Fehlerabschätzung für das Differenzenverfahren zur Lösung partieller Differentialgleichungen”, Z. Angew. Math. Mech. 10 (1930), 373–382.CrossRefGoogle Scholar
[21]Jamet, Pierre, “Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain”, SIAM J. Numer. Anal. 15 (1978), 912–928.CrossRefGoogle Scholar
[22]Jespersen, Dennis, “Ritz-Galerkin methods for singular boundary value problems” SIAM J. Numer. Anal. 15 (1978), 813–834.CrossRefGoogle Scholar
[23]Johnson, C., Nävert, U. and Pitkäranta, J., “Finite element methods for linear hyperbolic problems”, Comput. Methods Appl, Mech. Engrg. (to appear).Google Scholar
[24]Lasaint, P. and Raviart, P.A., “On a finite element method for solving the neutron transport equation”, Mathematical aspects of finite elements in partial differential equations, 89–123(Proc. Sympos. Math. Res. Center,University of Wisconsin,Madison,1974, Publications No. 33. Academic Press, New york, London, 1974).CrossRefGoogle Scholar
[25]Natterer, Frank, “Über die punktweise Knovergenz finiter Elemente”, Numer. Math. 25 (1975), 67–77.CrossRefGoogle Scholar
[26]Nitsche, J., “Lineare Spline-Funktionen und die Methoden von Ritz für elliptische Ranndwertprobleme”, Arch. Rational Mech. Anal. 36 (1970), 348–355.CrossRefGoogle Scholar
[27]Nitsche, J., “L∞-convergence of finite element approximation”,Second conference on finite elements,(Rennes, France,1975).Google Scholar
[28] Л.А. Оганесян [Oganesjan, L.A.], “Сходимость разностных схеь при улучненной aппроксиачии гранипы” [Convergence of difference schemes in case of improved approximation of the bountary], Ž. Vyčisl. Mat i Mat. Fiz. 6 (1966), 1029–1042.Google Scholar
[29] Л.А. Оганесян, Л.А. Руховец [Oganesjan, L.A., Rukhovets, L.A.], “Иследование скорорти сходимости вариационо–разностных схем дцяэппиптических уравнений иторого порялка в лвумрной облкφсти сгладкой границей” [Study of the rate of convergence of variational difference schemes for second order elliptic equations in two-dimensional regions with smooth boundaries”, Ž. Vyčisl. Mat i Mat. Fiz. 9 (1969), 1102–1120.Google Scholar
[30]Raviart, Pierre-Arnaud, “The use of numerical integration in finite element methods for solving parabolic equations”, Topics in numerical analysis, 233–264 (Proc. Roy. Irish Acad. Conf.,University College,Dublin,1972. Academic Press, New York, London, 1973).Google Scholar
[31]Rayleigh, Lord, Theory of sound, Volume I, Second Edition (Macmillan, London, 1894).Google Scholar
[32]Rayleigh, Lord, Theory of sound, Volume II, Second Edition (Macmillan, London, 1896).Google Scholar
[33]Richtmyer, Robert D., Difference methode for initial-value problems (John Wiley & Sons, New York, London, 1957).Google Scholar
[34]Ritz, Walter, “Über eine neue Methode zur Lösung gewisser Variations-probleme der Mathematischen Physik” J. Reine Angew. Math. 135 (1908), 1–61.Google Scholar
[35]Schreiber, Robert and Eisenstat, Stanley C., “Finite element methods for spherically symmetric elliptic equations” SIAM J. Numer. Anal. 18 (1981), 546–558.CrossRefGoogle Scholar
[36]Scott, Ridgway, “Optimal L ∞ estimates for the finite element method on irregular meshes”, Math. Comp. 30 (1976), 681–697.Google Scholar
[37]Strang, Gilbert and Fix, George J.An analysis of the finite element method (Prentice-Hall, Englewood Cliffs, New Jersey, 1973).Google Scholar
[38]Thomée, Vidar, “Spline approximation and difference schemes for the heat equation”, The mathematical foundations of the finite element method with applications in partial differential equations, 711–746 (Proc. Sympos.University of Maryland,Baltimore, Maryland,1972. Academic Press, New York, London, 1973).CrossRefGoogle Scholar
[39]Thomée, Vidar, “Galerkin-finite element methods for parabolic equations”,Proceedings of the International Congress of Mathematicians,Helsinki,1978,943–592 (Acad. Sci, Fennica, Helsinki, 1980).Google Scholar
[40]Turner, M.J., Clough, R.W., Martin, H.C. and Topp, L., “Stiffness and deflection analysis of complex structures”, J. Aero. Sci. 23 (1956), 805–823.CrossRefGoogle Scholar
[41]Ushijima, Teruo, “Error estimates for the lumped mass approximation of the heat equation”, Mem. Numer. Math. 6 (1979), 65–82.Google Scholar
[42]Zienkiewicz, O.C., The finite element method in engineering science, Third Edition (McGraw-Hill, London, New York, 1977).Google Scholar
[43]Zienkiewicz, O.C., and Cheung, Y.K., “Finite elements in the solution of field problems”, Engineer 220 (1965), 507–510.Google Scholar
[44]Zlámal, Miloš, “On the finite element method” Numer. Math. 12 (1968), 394–409.CrossRefGoogle Scholar