Backward Euler method for abstract time-dependent parabolic equations with variable domains

摘要

We consider semidiscretizations in time, based on the backward Euler method, of an abstract, non-autonomous parabolic initial value problem u'(t)=A(t)u(t); u(0)=u_0, where A(t):D(A(t)) is contained in X -> X, 0<=t<=T, is a family of sectorial operators in a Banach space X. The domains D(A(t)) are allowed to depend on t. Our hypotheses are fulfilled for classical parabolic problems in the L~p, 1<+infinity, norms. We prove that the semidiscretization is stable in a suitable sense. We get optimal estimates for the error even when non-homogeneous boundary values are considered. In particular, the results are applicable to the analysis of the semidiscretizations of time-dependent parabolic problems under non-homogeneous Neumann boundary conditions.