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

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Pressure-robust and conforming

discretization of the Stokes equations on

anisotropic meshes

Volker Kempf

October 7, 2022

Abstract

Pressure-robust discretizations for incompressible ﬂows have been in the focus of

research for the past years. Many publications construct exactly divergence-free methods or use a

reconstruction approach [13] for existing methods like the Crouzeix–Raviart element in order to

achieve pressure-robustness. To the best of our knowledge, except for our recent publications [3, 4],

all those articles impose a condition on the shape-regularity of the mesh, and the two mentioned

papers that allow for anisotropic elements use a non-conforming velocity approximation. Based

on the classical Bernardi–Raugel element we provide a conforming pressure-robust discretization

using the reconstruction approach on anisotropic meshes. Numerical examples support the theory.

1 Introduction

During the last years, pressure-robustness has emerged as an important property that

discretizations for incompressible ﬂow problems should possess. For the Stokes problem

in a domain Ω that for a data function f∈L2(Ω) and viscosity ν > 0 is given by

−ν∆u+∇p=fin Ω,(1a)

∇ · u= 0 in Ω,(1b)

a pressure-robust method yields velocity error estimates of the form, see [13],

ku−uhk1,h .inf

vh∈Xhku−vhk1,h +hm|u|m+1,

where

is the discrete velocity space and

kvk2

1,h

PT∈Thk∇vk2

0,T

. Missing pressure-

robustness on the other hand, e.g., in the case of the classical family of Taylor–Hood

elements, leads to error estimates of the type

ku−uhk1.inf

vh∈Xhku−vhk1+1

νinf

qh∈Qhkp−qhk0,

arXiv:2210.02756v1 [math.NA] 6 Oct 2022

where

is the discrete pressure space. Both estimates contain the best-approximation

error for the velocity in the discrete velocity space, however the advantage of the ﬁrst

estimate is obvious and leads to the descriptive name pressure-robust: the velocity error

does not depend on the pressure approximability and the viscosity of the ﬂuid.

Due to intensive research, many pressure-robust methods are known, e.g., the Scott–

Vogelius element [15],

(

div

)-conforming discontinuous Galerkin methods [7, 12] or

classical methods using a reconstruction approach to gain pressure-robustness [13]. The

proofs for all of these methods however rely on the assumption of shape-regularity on the

mesh elements, which excludes anisotropically graded meshes for boundary layers or edge

singularities, which may occur in ﬂow problems. This shortcoming was treated in our

publications [3, 4], where the pressure-robust variant of the Crouzeix–Raviart method

was used and we could show error estimates for anisotropic meshes in the boundary layer

and edge singularity settings.

Since the velocity approximation of the Crouzeix–Raviart method is non-conforming,

the aim of this contribution is to present a pressure-robust and conforming method which

can be used for meshes that contain anisotropic elements. The presented theory of this

paper is contained in [11] in a more abstract setting.

2 Reconstruction approach for pressure-robustness

In order to achieve pressure-robustness, we employ the reconstruction approach introduced

in [13]. Consider problem

(1)

on a domain Ω

⊂R2

with viscosity parameter

ν >

0 and

homogeneous Dirichlet boundary conditions. The weak form of this problem is well known:

Find (u, p)∈X×Q=H1

0(Ω) ×L2

0(Ω) so that

ν(∇u,∇v)−(∇ · v, p)−(∇ · u, q)=(f,v)∀(v, q)∈X×Q. (2)

Since we later require that for the solution (

u, p

)

∈H2

(Ω)

×H1

(Ω) holds, we assume

that Ω is a convex polygon where this required regularity is guaranteed [10].

By using the Helmholtz–Hodge decomposition of the data

∇φ

into a

divergence-free part

and an irrotational part

∇φ

, and looking at the problem in the

subspace of divergence free functions X0={v∈X: (∇ · v, q)=0∀q∈Q}

Find u∈X0so that ν(∇u,∇v) = (f,v) = (Pf,v)∀v∈X0,(3)

we see that the velocity solution is independent of the gradient part

∇φ

of the data, see

[13], as the test functions from

are

-orthogonal on gradients. We aim to preserve

this property in the discrete setting by using a reconstruction operator

, see [13], on

the velocity test functions on the right hand side of the problem, so that the discrete

version of (2) is given by

νah(uh,vh) + bh(vh, ph) + bh(uh, qh)=(f, Ihvh)∀(vh, qh)∈Xh×Qh,(4)

where

(

uh,vh

) = (

∇uh,∇vh

) and

(

vh, ph

) =

−

(

∇ · vh, ph

). Similar to

(3)

we can

write this problem in the subspace of discretely divergence-free functions

{vh∈

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摘要：

Pressure-robustandconformingdiscretizationoftheStokesequationsonanisotropicmeshesVolkerKempfOctober7,2022AbstractPressure-robustdiscretizationsforincompressibleowshavebeeninthefocusofresearchforthepastyears.Manypublicationsconstructexactlydivergence-freemethodsoruseareconstructionapproach[13]forexis...

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

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