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

We consider semidiscretizations in time, based on the backward Euler method, of an abstract, non-autonomous parabolic initial value problem where , , is a family of sectorial operators in a Banach space X. The domains are allowed to depend on t. Our hypotheses are fulfilled for classical parabolic problems in the , , 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. Received October 17, 1997 / Revised version received April 17, 1998