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

Abstract

Recently, the $\vec{H}(\mathrm{div})$-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.

References in corpus (2)

• Divergence-free $H$(div)-FEM for time-dependent incompressible flows with applications to high Reynolds number vortex dynamics
• Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier-Stokes equations

Cited by in corpus (7)

• A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes
• Pressure-robust error estimate of optimal order for the Stokes equations on domains with edges
• The Scott-Vogelius Method for Stokes Problem on Anisotropic Meshes
• Brezzi--Douglas--Marini interpolation on anisotropic simplices and prisms
• Pressure-robustness for the Stokes equations on anisotropic meshes
• Anisotropic $H_{div}$-norm error estimates for rectangular $H_{div}$-elements
• Pressure-robust and conforming discretization of the Stokes equations on anisotropic meshes