A Family of H(div) Finite Element Approximations on Polygonal Meshes

Abstract

In this paper, we present a family of $H(\div)$-compatible finite element spaces on strictly convex $n$-gons, whose construction makes use of generalized barycentric coordinates. In particular, for integers $0\leq k\leq2$, we define finite element spaces with edge degrees of freedom that include polynomial vector fields of order $k$ and whose vector fields have piecewise $k$th-order polynomial normal traces along the element boundary. These spaces suffer from the shortcoming that the image of the divergence operator includes nonpolynomial functions and, as such, their direct use in a mixed setting along with polynomial scalar fields may lead to unstable discretizations exhibiting degraded or no convergence. We present a general remedy for restoration of optimal convergence that involves “polynomial corrections” of vector fields and their divergence at the element level. These corrections are consistent with one another and require computation of suitable polynomial projection maps. In addition to the theo...