Space-time error estimates for approximations of linear parabolic problems with generalized time boundary conditions

We first give a general error estimate for the nonconforming approximation of a problem for which a Banach-Ne{\v c}as-Babu{\v s}ka (BNB) inequality holds. This framework covers parabolic problems with general conditions in time (initial value problems as well as periodic problems) under minimal regularity assumptions. We consider approximations by two types of space-time discretizations, both based on a conforming Galerkin method in space. The first one is the Euler $\theta$--scheme. In this case, we show that the BNB inequality is always satisfied, and may require an extra condition on the time step for $\theta$ $\le$ 1 2. The second one is the time discontinuous Galerkin method, where the BNB condition holds without any additional condition.