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Superconvergence analysis of flux computations for nonlinear problems

Published online by Cambridge University Press:  17 April 2009

S.-S. Chow  and
R.D. Lazarov
S.-S. Chow
Affiliation:
Department of MathematicsUniversity of WyomingLaramie, United States of America
R.D. Lazarov
Affiliation:
Bulgarian Academy of SciencesInstitute of MathematicsSofia, Bulgaria
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Abstract

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In this paper we consider the error estimates for some boundary-flux calculation procedures applied to two-point semilinear and strongly nonlinear elliptic boundary value problems. The case of semilinear parabolic problems is also studied. We prove that the computed flux is superconvergent with second and third order of convergence for linear and quadratic elements respectively. Corresponding estimates for higher order elements may also be obtained by following the general line of argument.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 3 , December 1989 , pp. 465 - 479
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Andreev, A.B. and Lazarov, R.D., ‘Superconvergence of the gradient for quadratic triangular finite elements’, Numer. Methods, Partial Differential Equation, 4 (1988), 15–31.CrossRefGoogle Scholar
[2]Barrett, J.W. and Elliott, C.M., ‘Total flux estimates for a finite–element approximation of elliptic equations’, I.M.A. J. Numer. Anal. 7 (1987), 129–148.Google Scholar
[3]Browder, F.E., ‘Existence and uniqueness theorems for solutions of nonlinear boundary value problems’, Proc. Sympos. Appl. Math. 17 (1065), 24–49.Google Scholar
[4]Carey, C.F., ‘Derivative calculations from finite element solutions’, J. Comp. Meth. Applied Mech. Eng. 35 (1982), 1–4.Google Scholar
[5]Carey, G.F., Chow, S.-S. and Seager, M., ‘Approximate boundary–flux calculations’, J. Comp. Meth. Applied Mech. Eng. 50 (1985), 107–120.Google Scholar
[6]Chow, S.-S., ‘Finite element error estimates for non-linear elliptic equations of monotone type’, Numer. Math. 54 (1989), 373–393.CrossRefGoogle Scholar
[7]Ciarlet, P.E., The finite element method for elliptic problems (North–Holland, Amsterdam, 1978).Google Scholar
[8]Douglas, J. Jr. and Dupont, T., ‘Galerkin approximations for the two point boundary problems using continuous piecewise polynomial spaces’, Numer. Math. 22 (1974), 99–109.Google Scholar
[9]Douglas, J., Dupont, T. and Wheeler, M.F., ‘A Galerkin procedure for approximating the flux on the boundary for elliptic and parabolic boundary value problems’, RAIRO Anal. Numer. 2 (1974), 47–59.Google Scholar
[10]Dupont, T., ‘A unified theory of superconvergence for Galerkin methods for two–point boundary problems’, SIAM J. Numer. Anal. 13 (1976), 362–368.CrossRefGoogle Scholar
[11]Glowiniski, R. and Marrocco, A., ‘Analyse numérique du champ magnétique d'un alternateur par élements finis et sur-relaxation ponctuelle non liéaire’, J. Comp. Meth. Applied Mech. Eng. 3 (1974), 55–85.Google Scholar
[12]Lazarov, R.D., Pehlivanov, A.I., Chow, S.-S. and Carey, G.F., ‘Superconvergence analysis of boundary-flux calculations’, (in preparation).Google Scholar
[13]Křižek, M. and Neittaanmäki, P., ‘On superconvergence techniques’, Acta. Appl. Math. 9 (1987), 175–198.Google Scholar
[14]Noor, M.A. and Whiteman, J.R., ‘Error bounds for finite element solutions of mildly nonlinear elliptic boundary value problems’, Numer. Math. 26 (1976), 107–116.Google Scholar
[15]Strang, C. and Fix, G.J., An analysis of the finite element method (Prentice-Hall, Englewood Cliffs, 1971).Google Scholar
[16]Thomée, V., Gaterkin finite element methods for parabolic problems: Lecture Notes in Math. 1054 (Springer-Verlag, Berlin, 1984).Google Scholar
[17]Vainberg, M.M., Variational method and method of monotone operators in the theory of nonlinear equations (Halsted Press, John Wiley, New York, 1973).Google Scholar
[18]Wheeler, J.A., ‘Simulation of heat transfer from a warm pipeline buried in permafrost’, 74th National Meeting of American institute of Chemical Engineers, New Orleans.Google Scholar
[19]Wheeler, M.F., ‘A priori L 2 error estinmates for Galerton approximations to parabolic partial differential equations’, SIAM J. Numer. Anal. 10 (1973), 723–759.Google Scholar
[20]Wheeler, M.F., ‘A Galerkin procedure for estimating the flux for two-point boundary value problems’, SIAM J. Numer. Anal. 11 (1974), 764–768.Google Scholar
[21]Alámal, M., ‘Some superconvergence results in the finite element methods’ in Mathematical Aspects of Finite Element Methods: Lecture Notes in Math. 606, pp. 363–362 (Springer-Verlag, Berlin, 1977).Google Scholar