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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 useH−1H^{-1}loads. We prove that any boundedH−1H^{-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.

On a Mixed FEM and a FOSLS with 𝐻 −1 Loads