We study a nonstandard 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 $\textbf{H}(\operatorname{\textrm{div}};\varOmega )$ 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.

中文翻译：

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关于混合有限元公式 inf-sup 条件的必要性

我们研究泊松问题的非标准混合公式，有时称为对偶混合公式。出于与通量平衡相关的原因，我们使用符合 $\textbf{H}(\operatorname{\textrm{div}};\varOmega )$ 的有限元来近似梯度，即使公式将允许不连续的有限元。该方案不是一致 inf-sup 稳定的，但我们可以证明解的存在性和唯一性，以及在做出适当的正则性假设时对梯度变量的最优误差估计。本文还补充了一些补充说明，阐明了混合配方不稳定的根源。