Sharp anisotropic interpolation error estimates for rectangular Raviart-Thomas elements

Abstract:

Anisotropic $L_p$-norm error estimates are derived for the standard rectangular Raviart-Thomas elements $RT_{[k]}(\tilde K)$ in $\mathbb {R}^d$ for $p\in [1, \infty ],\ k\ge 0$ and $d \ge 2$. Here $\tilde K$ is an affine image of an axi-parallel parallelotope $K$. The proofs are based on a variant of the classical Poincaré inequality. The estimates derived make full use of the asymmetric nature of the vector space components of $RT_{[k]}(\tilde K)$; a Shishkin mesh example demonstrates their superiority over previous estimates.