Under some regularity assumptions, we report an a priori error analysis of a dG scheme for the Poisson and Stokes flow problem in their dual mixed formulation. Both formulations satisfy a Babu\v{s}ka-Brezzi type condition within the space H(div) x L2. It is well known that the lowest order Crouzeix-Raviart element paired with piecewise constants satisfies such a condition on (broken) H1 x L2 spaces. In the present article, we use this pair. The continuity of the normal component is weakly imposed by penalizing jumps of the broken H(div) component. For the resulting methods, we prove well-posedness and convergence with constants independent of data and mesh size. We report error estimates in the methods natural norms and optimal local error estimates for the divergence error. In fact, our finite element solution shares for each triangle one DOF with the CR interpolant and the divergence is locally the best-approximation for any regularity. Numerical experiments support the findings and suggest that the other errors converge optimally even for the lowest regularity solutions and a crack-problem, as long as the crack is resolved by the mesh.

翻译：在一定的正则性假设下，我们针对泊松方程和斯托克斯流问题在对偶混合公式中的dG格式进行了先验误差分析。这两种公式均在空间H(div)×L2中满足Babuška-Brezzi型条件。众所周知，最低阶Crouzeix-Raviart元与分片常数配对在（分裂）H1×L2空间中满足该条件。本文采用这一配对。通过惩罚分裂H(div)分量的跳跃，弱施加法向分量的连续性。对于所得方法，我们证明了适定性和收敛性，其常数独立于数据和网格尺寸。我们给出了方法自然范数下的误差估计，以及散度误差的最优局部误差估计。实际上，对于每个三角形，我们的有限元解与CR插值函数共享一个自由度，且无论正则性如何，散度均为局部最佳逼近。数值实验支持这些结论，并表明即使对于最低正则解和裂纹问题（只要网格解析裂纹），其他误差也能达到最优收敛。