Semidiscretization in Time for Parabolic Problems
نویسنده
چکیده
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 fact strongly ^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). I. Semidiscretization in Time by a Single-Step Method.
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تاریخ انتشار 2010