We study a non standard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in H(div) for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup stable, but we can show existence and uniqueness of the solution, as well as optimal error estimates for the gradient variable when suitable regularity assumptions are made. Several additional remarks complete the paper, shedding some light on the sources of instability for mixed formulations.

翻译：我们研究普瓦松问题的非标准混合配方,有时称为双重混合配方,出于平衡通量的原因,我们使用H(div)中符合的有限元素来接近梯度,即使配方允许不连续的有限元素。这个办法并不统一稳定,但我们可以显示解决办法的存在和独特性,以及当提出适当的规律性假设时,对梯度变量的最佳误差估计。一些补充意见完成了文件,对混合配方的不稳定来源作了一些说明。