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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>H</m:mi> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> H⁻¹ loads. We prove that any bounded <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>H</m:mi> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> 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.