Anisotropic error estimates for the $L^2$-projection onto discontinuous finite element spaces

Abstract

Anisotropic error estimates are determined for the $L^2$-orthogonal projection onto discontinuous finite element spaces in two and three dimensions. These estimates are determined in $W^{m,p}$-seminorms, in which $p\in [1,\infty ]$ and $0\le m\le k+1$, where $k\ge 0$ is the polynomial degree. The analysis relies on a geometric condition that is equivalent to the maximum angle condition. As an application, an apriori error analysis is presented for a hybridizable discontinuous Galerkin discretization of the Poisson problem on two and three dimensional anisotropic meshes.