DeepAI AI Chat
Log In Sign Up

Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem

by   Derk Frerichs, et al.

Non divergence-free discretisations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterised by large discretisations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods (VEM) on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretised in a careful manner. To be able to evaluate the right-hand side for the test functions, some explicit interpolation of the virtual test functions is needed that can be evaluated pointwise everywhere. The standard discretisation via an L^2-best approximation does not preserve the divergence and so destroys the orthogonality between divergence-free test functions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness another divergence-preserving reconstruction is suggested based on Raviart-Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes.


page 1

page 2

page 3

page 4


A formal construction of a divergence-free basis in the nonconforming virtual element method for the Stokes problem

We develop a formal construction of a pointwise divergence-free basis in...

Pressure-robustness in the context of optimal control

This paper studies the benefits of pressure-robust discretizations in th...

Pressure-robust and conforming discretization of the Stokes equations on anisotropic meshes

Pressure-robust discretizations for incompressible flows have been in th...

A compact divergence-free H(div)-conforming finite element method for Stokes flows

In this paper, we construct a P_1^c⊕ RT0-P0 discretization of the Stokes...

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

This paper contains two major contributions. First we derive, following ...