On the discretization in time of semilinear parabolic equations with nonsmooth initial data

Authors:
Michel Crouzeix and Vidar Thomée

Journal:
Math. Comp. **49** (1987), 359-377

MSC:
Primary 65M10

DOI:
https://doi.org/10.1090/S0025-5718-1987-0906176-3

MathSciNet review:
906176

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Abstract: Single-step discretization methods are considered for equations of the form , where *A* is a linear positive definite operator in a Hilbert space *H*. It is shown that if the method is consistent with the differential equation then the convergence is essentially of first order in the stepsize, even if the initial data *v* are only in *H*, but also that, in contrast to the situation in the linear homogeneous case, higher-order convergence is not possible in general without further assumptions on *v*.

**[1]**G. A. Baker, J. H. Bramble & V. Thomée, "Single step Galerkin approximations for parabolic problems,"*Math. Comp.*, v. 31, 1977, pp. 818-847. MR**0448947 (56:7252)****[2]**M. Crouzeix,*Sur l'Approximation des Équations Différentielles Opérationnelles Linéaires par des Méthodes de Runge-Kutta*, Thèse, Université Paris VI, 1975.**[3]**C. Johnson, S. Larsson, V. Thomée & L. B. Wahlbin, "Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data,"*Math. Comp.*, v. 49, 1987, pp. 331-357. MR**906175 (88k:65100)****[4]**A. H. Schatz & L. B. Wahlbin, "On the quasi-optimality in of the -projection into finite element spaces,"*Math. Comp.*v. 38, 1982, pp. 1-22. MR**637283 (82m:65106)**

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DOI:
https://doi.org/10.1090/S0025-5718-1987-0906176-3

Article copyright:
© Copyright 1987
American Mathematical Society