In this paper, we consider a fully-discrete approximation of an abstract evolution equation deploying a non-conforming spatial approximation and finite differences in time (Rothe–Galerkin method). The main result is the convergence of the discrete solutions to a weak solution of the continuous problem. Therefore, the result can be interpreted either as a justification of the numerical method or as an alternative way of constructing weak solutions. We formulate the problem in the very general and abstract setting of so-called non-conforming Bochner pseudo-monotone operators, which allows for a unified treatment of several evolution problems. Our abstract results for non-conforming Bochner pseudo-monotone operators allow to establish (weak) convergence just by verifying a few natural assumptions on the operators time-by-time and on the discretization spaces. Hence, applications and extensions to several other evolution problems can be performed easily. We exemplify the applicability of our approach on several DG schemes for the unsteady $p$-Navier–Stokes problem. The results of some numerical experiments are reported in the final section.

在本文中，我们考虑了一个抽象演化方程的完全离散近似，该方程采用了不一致的空间近似和时间有限差分（Rothe-Galerkin 方法）。主要结果是离散解收敛到连续问题的弱解。因此，结果可以解释为数值方法的合理性或构造弱解的替代方法。我们在所谓的不符合 Bochner 伪单调算子的非常一般和抽象的设置中制定问题，这允许统一处理几个进化问题。我们对不一致的 Bochner 伪单调算子的抽象结果允许仅通过逐时验证对算子和离散化空间的一些自然假设来建立（弱）收敛。因此，可以轻松执行对其他几个进化问题的应用和扩展。我们举例说明了我们的方法对不稳定的几个 DG 方案的适用性$p$-Navier-Stokes 问题。最后一节报告了一些数值实验的结果。