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Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data


Authors: Claes Johnson, Stig Larsson, Vidar Thomée and Lars B. Wahlbin
Journal: Math. Comp. 49 (1987), 331-357
MSC: Primary 65N10; Secondary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1987-0906175-1
MathSciNet review: 906175
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Abstract: We consider time-continuous spatially discrete approximations by the Galerkin finite element method of initial-boundary value problems for semilinear parabolic equations with nonsmooth or incompatible initial data. We find that the numerical solution enjoys a gain in accuracy at positive time of essentially two orders relative to the initial regularity, as a result of the smoothing property of the parabolic evolution operator. For higher-order elements the restriction to two orders is in contrast to known optimal order results in the linear case.


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  • [1] H. Amann, "Existence and stability of solutions for semi-linear parabolic systems, and applications to some diffusion reaction equations," Proc. Roy. Soc. Edinburgh Sect. A, v. 81, 1978, pp. 35-47. MR 529375 (80b:35078)
  • [2] J. J. Blair, Approximate Solution of Elliptic and Parabolic Boundary Value Problems, Thesis, University of California, Berkeley, 1970.
  • [3] J. H. Bramble, "A survey of some finite element methods proposed for treating the Dirichlet problem," Adv. in Math., v. 16, 1975, pp. 187-196. MR 0381348 (52:2245)
  • [4] J. H. Bramble, A. H. Schatz, V. Thomée & L. B. Wahlbin, "Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations," SIAM J. Numer. Anal., v. 14, 1977, pp. 218-241. MR 0448926 (56:7231)
  • [5] M. Crouzeix & V. Thomée, On the Discretization in Time of Semilinear Parabolic Equations with Nonsmooth Initial Data, Université de Rennes, 1985. (Preprint.) MR 906176 (89c:65102)
  • [6] H. Fujita & A. Mizutani, "On the finite element method for parabolic equations, I: Approximation of holomorphic semigroups," J. Math. Soc. Japan, v. 28, 1976, pp. 749-771. MR 0428733 (55:1753)
  • [7] J. K. Hale, X.-B. Lin & G. Raugel, "Upper semicontinuity of attractors for approximations of semigroups and partial differential equations," Math. Comp. (To appear.) MR 917820 (89a:47093)
  • [8] R. Haverkamp, Eine Aussage zur $ {L_\infty }$-Stabilität und zur genauen Konvergenzordnung der $ H_0^1$-Projektionen, Preprint no. 508, Universität Bonn, 1982.
  • [9] H.-P. Helfrich, "Fehlerabschätzungen für das Galerkinverfahren zur Lösung von Evolutionsgleichungen," Manuscripta Math., v. 13, 1974, pp. 219-235. MR 0356513 (50:8983)
  • [10] J. G. Heywood & R. Rannacher, "Finite element approximation of the nonstationary Navier-Stokes problem. Part III. Smoothing property and higher order error estimates for spatial discretization," SIAM J. Numer. Anal. (To appear.) MR 942204 (89k:65114)
  • [11] O. A. Ladyženskaja, V. A. Solonnikov & N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Transl. Math. Monographs, Vol. 23, Amer. Math. Soc., Providence, R. I., 1968.
  • [12] S. Larsson, On Reaction-Diffusion Equations and Their Approximation by Finite Element Methods, Thesis, Chalmers University of Technology, Göteborg, Sweden, 1985.
  • [13] J. Moser, "A rapidly convergent iteration method and nonlinear partial differential equations. I," Ann. Scuola Norm. Sup. Pisa, v. 20, 1966, pp. 265-315. MR 0199523 (33:7667)
  • [14] J. A. Nitsche, "Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind," Abh. Math. Sem. Univ. Hamburg, v. 36, 1971, pp. 9-15. MR 0341903 (49:6649)
  • [15] J. A. Nitsche, $ {L_\infty }$-Convergence of Finite Element Approximation, 2. Conference on Finite Elements (Rennes, 1975), Univ. Rennes, Rennes, 1975. MR 568857 (81e:65058)
  • [16] J. A. Nitsche & M. F. Wheeler, " $ {L_\infty }$-boundedness of the finite element Galerkin operator for parabolic problems," Numer. Funct. Anal. Optim., v. 4, 1981/82, pp. 325-353. MR 673316 (84a:65083)
  • [17] R. Rannacher & R. Scott, "Some optimal error estimates for piecewise linear finite element approximations," Math. Comp., v. 38, 1982, pp. 437-445. MR 645661 (83e:65180)
  • [18] A. H. Schatz & L. B. Wahlbin, "On the quasi-optimality in $ {L_\infty }$ of the $ \dot{H}^1$-projection into finite element spaces," Math. Comp., v. 38, 1982, pp. 1-22. MR 637283 (82m:65106)
  • [19] A. H. Schatz, V. Thomée & L. B. Wahlbin, "Maximum norm stability and error estimates in parabolic finite element equations," Comm. Pure Appl. Math., v. 33, 1980, pp. 265-304. MR 562737 (81g:65136)
  • [20] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, Berlin and New York, 1983. MR 688146 (84d:35002)
  • [21] G. Strang & G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, N. J., 1973. MR 0443377 (56:1747)
  • [22] V. Thomée, "Negative norm estimates and superconvergence in Galerkin methods for parabolic problems," Math. Comp., v. 34, 1980, pp. 93-113. MR 551292 (81a:65092)
  • [23] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin and New York, 1984.
  • [24] V. Thomée & L. B. Wahlbin, "On Galerkin methods in semilinear parabolic problems," SIAM J. Numer. Anal., v. 12, 1975, pp. 378-389. MR 0395269 (52:16066)
  • [25] L. B. Wahlbin, "A remark on parabolic smoothing and the finite element method," SIAM J. Numer. Anal., v. 17, 1980, pp. 33-38. MR 559459 (81k:65128)
  • [26] M. F. Wheeler, "A priori $ {L_2}$ error estimates for Galerkin approximations to parabolic partial differential equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 723-759. MR 0351124 (50:3613)

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DOI: https://doi.org/10.1090/S0025-5718-1987-0906175-1
Article copyright: © Copyright 1987 American Mathematical Society

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