We study the error to the discretization in time of a parabolic evolution equation by a single-step method or by a multistep method when the initial condition is not regular.Introduction. The problem we are considering is the parabolic evolution equation ( u'(t) + Au(t) = 0, 0 3 is documented in [8] and [2]. It is shown in [8] that for p > 3, rp is in factstrongly ^4(öp)-stable for some 0 < 8p < n/2. For small p, 6 is close to 7r/2 and in the special cases p = 3, 4, r is A -stable. Examples of rational approximations to e~z which are strongly A(6)-stable with r(°°) = 0 are provided by the family rv(z) developed in [2].In the second part, we investigate error estimates when the discretization in time is carried out by means of a multistep method. Zlamal gives an error bound under the assumption that the operator A is selfadjoint and the method strongly ^4(0)-stable.Here, error estimates are obtained if the operator A is maximal sectorial and the method strongly ,4(0)-stable (0 < 0 < rr/2).

In this paper the author considers the first boundary value problem for the nonlinear equation ut Aum aum in f, a smooth bounded domain in ]n with the zero lateral boundary condition and with a positive initial condition; m is supposed to be larger than one and a positive.A scheme for the discretization in time of that problem is proposed. It is proven that if the exact solution blows up in a finite time, it is the same for the numerical solution. Estimates of the blow-up time are obtained. The stability of the method and the convergence for a class of initial conditions is proved.

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Abstract. In this paper, the authors consider the first boundary value problem for the nonlinear reaction diffusion equation: ut − ∆u m = αu p 1 in Ω, a smooth bounded domain in R d (d ≥ 1) with the zero lateral boundary condition and with a positive initial condition, m ∈ ]0, 1[ (fast diffusion problem), α ≥ 0 and p 1 ≥ m. Sufficient conditions on the initial data are obtained for the solution to vanish or become infinite in a finite time. A scheme for the discretization in time of this problem is proposed. The numerical scheme preserves the essential properties of the initial problem; namely existence of an extinction or a blow-up time, for which estimates have been obtained. The convergence of the method is also proved.

We study the error due to the discretization in time of a nonlinear parabolic problem by a multistep method. Error estimates are obtained if the method is of the order p (p> 1) and strongly A(O)-stable 0