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.

中文翻译：

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非一致性对偶混合方法的 Crouzeix-Raviart 元素：先验分析

在一些规律性假设下，我们报告了对双混合公式中泊松和斯托克斯流问题的 dG 方案的先验误差分析。两种公式都满足空间 H(div) x L2 内的 Babu\v{s}ka-Brezzi 类型条件。众所周知，与分段常数配对的最低阶 Crouzeix-Raviart 元素在（破碎的）H1 x L2 空间上满足这样的条件。在本文中，我们使用这对。法线分量的连续性是通过惩罚损坏的 H(div) 分量的跳跃而弱强加的。对于由此产生的方法，我们证明了与数据和网格大小无关的常数的适定性和收敛性。我们报告了自然范数方法中的误差估计和散度误差的最佳局部误差估计。实际上，我们的有限元解决方案为每个三角形共享一个 DOF 与 CR 插值，并且散度是任何规则性的局部最佳近似值。数值实验支持这些发现，并表明即使对于最低规律性解决方案和裂缝问题，只要裂缝由网格解决，其他误差也能最佳收敛。