Arbitrary-order pressure-robust DDR and VEM methods for the Stokes problem on polyhedral meshes

TL;DR

This paper introduces pressure-robust DDR and VEM methods for the Stokes problem on polyhedral meshes, achieving high-order accuracy without submesh conforming, and explores their theoretical relations and properties.

Contribution

The paper develops pressure-robust, high-order DDR and VEM methods for the Stokes equations on polyhedral meshes, avoiding submesh conforming constructions and analyzing their interrelations.

Findings

Pressure-robust error estimates of order h^{k+1} are proven.

Numerical tests confirm theoretical error estimates.

DDR and VEM complexes can be reformulated into each other, satisfying commuting properties.

Abstract

This paper contains two major contributions. First we derive, following the discrete de Rham (DDR) and Virtual Element (VEM) paradigms, pressure-robust methods for the Stokes equations that support arbitrary orders and polyhedral meshes. Unlike other methods presented in the literature, pressure-robustness is achieved here without resorting to an $\boldsymbol{H}({\rm div})$-conforming construction on a submesh, but rather projecting the volumetric force onto the discrete $\boldsymbol{H}({\bf curl})$ space. The cancellation of the pressure error contribution stems from key commutation properties of the underlying DDR and VEM complexes. The pressure-robust error estimates in $h^{k+1}$ (with $h$ denoting the meshsize and $k\ge 0$ the polynomial degree of the DDR or VEM complex) are proven theoretically and supported by a panel of three-dimensional numerical tests. The second major contribution of the paper is an in-depth study of the relations between the DDR and VEM approaches. We show, in particular, that a complex developed following one paradigm admits a reformulation in the other, and that couples of related DDR and VEM complexes satisfy commuting diagram properties with the degrees of freedom maps.