Brezzi-Douglas-Marini interpolation of any order on anisotropic triangles and tetrahedra

TL;DR

This paper extends optimal error estimates for Brezzi-Douglas-Marini finite element interpolation to anisotropic triangles and tetrahedra, crucial for accurate simulations in fluid dynamics with boundary layers.

Contribution

It provides the first optimal error estimates for BDM interpolation on anisotropic elements, relaxing standard angle conditions and including numerical validation.

Findings

Optimal error estimates are established for BDM interpolation on anisotropic elements.

The results relax the minimal angle condition typically required.

Numerical experiments on Stokes equations confirm theoretical predictions.

Abstract

Recently, the $\vec{H}(\operatorname{div})$-conforming finite element families for second order elliptic problems have come more into focus, since due to hybridization and subsequent advances in computational efficiency their use is no longer mainly theoretical. Their property of yielding exactly divergence-free solutions for mixed problems makes them interesting for a variety of applications, including incompressible fluids. In this area, boundary and interior layers are present, which demand the use of anisotropic elements. While for the Raviart-Thomas interpolation of any order on anisotropic tetrahedra optimal error estimates are known, this contribution extends these results to the Brezzi-Douglas-Marini finite elements. Optimal interpolation error estimates are proved under two different regularity conditions on the elements, which both relax the standard minimal angle condition. Additionally a numerical application on the Stokes equations is presented to illustrate the findings.