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.

翻译：我们首先给出一个一般性误差估计，适用于满足Banach-Nečas-Babuška（BNB）不等式的问题的非协调近似。该框架在最小正则性假设下，涵盖了具有一般时间条件的抛物型问题（包括初值问题以及周期问题）。我们考虑两类时空离散化的近似方法，两者均基于空间中的协调Galerkin方法。第一类是Euler $\theta$ 格式。在这种情况下，我们证明BNB不等式始终成立，但可能需要对时间步长施加额外条件，当$\theta \le 1/2$。第二类是时间非连续Galerkin方法，其中BNB条件无需任何附加条件即可成立。