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 H^{-1} loads. We prove that any bounded H − 1 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 Clément quasi-interpolator. We prove that this Clément 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.

中文翻译：

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在具有 𝐻−1 载荷的混合 FEM 和 FOSLS 上

我们研究泊松问题的混合有限元法（混合 FEM）和一阶系统最小二乘有限元 (FOSLS) 的变体，其中我们用允许使用的合适的正则化替换载荷 H − 1个 H^{-1} 负载。我们证明任何有界 H − 1个 H^{-1} 投影到分段常数上可用于定义正则化并产生最低阶混合 FEM 的准最优性。FOSLS 在较弱的规范中。给出了此类投影仪的构造示例。一种是基于加权 Clément 准插值器的伴随。我们证明了这个 Clément 算子具有二阶逼近特性。对于修改后的混合方法，我们展示了在最小正则假设下后处理解的最佳收敛速度——该结果对于没有正则化的最低阶混合 FEM 无效。数值例子总结了这项工作。