ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

15. Numerical Analysis and Scientific Computing

1991 MS Classification: 65 González, C., and & C. Palencia, Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Spain The implicit Euler method for abstract time-dependent parabolic problems with variable domains Let A(t) : D(A(t)) ⊂ X → X, 0 ≤ t ≤ T, be a family of linear operators in a Banach space X. For any individual t ∈ [0, T] the operator A(t) is assumed to generate an analytic semigroup e^{sA(t)}, s ≥ 0. The domains D(A(t)) are allowed to depend on 0 ≤ t ≤ T. Stability properties of the semidiscretizations in time of the initial value problem

u′(t) = A(t)u(t), 0 ≤ t ≤ T; u(0) = u_{0} ∈ X,

are studied. For simplicity, the attention is restricted to the case of the implicit Euler method. The problem is fitted in a new framework that affords for such a stability study and that covers many of the classical parabolic problems. Extrapolation spaces are not required in the present approach.