Sharp anisotropic interpolation error estimates for rectangular Raviart-Thomas elements

Abstract

Anisotropic L p L_p -norm error estimates are derived for the standard rectangular Raviart-Thomas elements R T [ k ] ( K ~ ) RT_{[k]}(\tilde K) in R d \mathbb {R}^d for p ∈ [ 1 , ∞ ] ,   k ≥ 0 p\in [1, \infty ],\ k\ge 0 and d ≥ 2 d \ge 2 . Here K ~ \tilde K is an affine image of an axi-parallel parallelotope K 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 R T [ k ] ( K ~ ) RT_{[k]}(\tilde K) ; a Shishkin mesh example demonstrates their superiority over previous estimates.