A Comparison of Finite Element Spaces for $H$(div) Conforming First-Order System Least Squares

Abstract

First-order system least squares (FOSLS) is a commonly used technique in a wide range of physical applications. FOSLS discretizations are straightforward to implement and offer many advantages over traditional Galerkin or saddle-point formulations. Often, these problems are formulated in $H(div)$ spaces and implemented using $H(div)$-conforming elements. These elements have fewer regularity assumptions than the more commonly used $H^1$-conforming elements and are therefore often believed to be more suited for less regular problems. This paper compares the approximation properties of the $H(div)$-conforming Raviart--Thomas and Brezzi--Douglas--Marini elements to $H^1$-conforming piecewise polynomials for $H(div)$ problem formulations. Furthermore, for each problem, a corresponding $H^1$-formulation is derived and compared. The traditional Poisson and Stokes problems are examined with less regular solutions addressed by adaptive refinement strategies. The results imply that for smooth problems or problems exhibiting a point singularity $H^1$-conforming piecewise polynomials are a viable, and sometimes more efficient, choice even if an $H(div)$-formulation is used. Smooth problems do additionally benefit by using an $H^1$-formulation instead of an $H(div)$-formulation.