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

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]**Garth A. Baker, James H. Bramble, and Vidar Thomée,*Single step Galerkin approximations for parabolic problems*, Math. Comp.**31**(1977), no. 140, 818–847. MR**0448947**, 10.1090/S0025-5718-1977-0448947-X**[2]**M. Crouzeix,*Sur l'Approximation des Équations Différentielles Opérationnelles Linéaires par des Méthodes de Runge-Kutta*, Thèse, Université Paris VI, 1975.**[3]**Claes Johnson, Stig Larsson, Vidar Thomée, and Lars B. Wahlbin,*Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data*, Math. Comp.**49**(1987), no. 180, 331–357. MR**906175**, 10.1090/S0025-5718-1987-0906175-1**[4]**A. H. Schatz and L. B. Wahlbin,*On the quasi-optimality in 𝐿_{∞} of the 𝐻¹-projection into finite element spaces*, Math. Comp.**38**(1982), no. 157, 1–22. MR**637283**, 10.1090/S0025-5718-1982-0637283-6

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

Article copyright:
© Copyright 1987
American Mathematical Society