We study variants of the mixed finite element method (mixed FEM) and the first-order system least-squares finite element (FOSLS) for the Poisson problem where we replace the load by a suitable regularization which permits to use $H^{-1}$ loads. We prove that any bounded $H^{-1}$ projector onto piecewise constants can be used to define the regularization and yields quasi-optimality of the lowest-order mixed FEM resp. FOSLS in weaker norms. Examples for the construction of such projectors are given. One is based on the adjoint of a weighted Cl\'ement quasi-interpolator. We prove that this Cl\'ement operator has second-order approximation properties. For the modified mixed method we show optimal convergence rates of a postprocessed solution under minimal regularity assumptions -- a result not valid for the lowest-order mixed FEM without regularization. Numerical examples conclude this work.

翻译：我们研究了泊松问题中混合有限元法（mixed FEM）和一阶系统最小二乘有限元法（FOSLS）的变体，其中通过合适的正则化替换载荷，从而允许使用$H^{-1}$载荷。我们证明，任何有界$H^{-1}$投影算子，当其作用于分片常数函数时，均可用于定义正则化，并在较弱的范数下实现最低阶混合有限元法或一阶系统最小二乘法的拟最优性。给出了此类投影算子的构造示例，其中一种基于加权的Clément拟插值算子的伴随算子。我们证明该Clément算子具有二阶逼近性质。对于修正的混合方法，我们在最小正则性假设下证明了后处理解的优化收敛速率——这一结果在无正则化的最低阶混合有限元法中并不成立。数值实验为本文收尾。