Anisotropic $H_{div}$-norm error estimates for rectangular $H_{div}$-elements

TL;DR

This paper derives sharp anisotropic $H_{div}$-norm error estimates for rectangular elements like Raviart-Thomas, Brezzi-Douglas-Marini, and Arnold-Boffi-Falk, crucial for convergence analysis in numerical methods.

Contribution

It provides the first anisotropic $H_{div}$-norm interpolation error estimates for key rectangular finite elements, enhancing convergence proofs.

Findings

Sharp anisotropic error estimates established

Applicable to Raviart-Thomas, BDM, and Arnold-Boffi-Falk elements

Supports improved convergence analysis in numerical simulations

Abstract

For the discretisation of $H_{div}$-functions on rectangular meshes there are at least three families of elements, namely Raviart-Thomas-, Brezzi-Douglas-Marini- and Arnold-Boffi-Falk-elements. In order to prove convergence of a numerical method using them, sharp interpolation error estimates are important. We provide them here in an anisotropic setting for the $H_{div}$-norm.