Divergence-Free HDG Methods for the Vorticity-Velocity Formulation of the Stokes Problem

Abstract Nondivergence-free discretizations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterized by large discretizations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretized 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 discretization 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.

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Divergence-free wavelet solution to the Stokes problem

Analysis in Theory and Applications ◽

10.1007/s10496-001-0083-3 ◽

2007 ◽

Vol 23 (1) ◽

pp. 83-91 ◽

Cited By ~ 2

Author(s):

Yingchun Jiang

Keyword(s):

Stokes Problem ◽

Divergence Free

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Spectral approximations of the Stokes problem by divergence-free functions

Journal of Scientific Computing ◽

10.1007/bf01061110 ◽

1987 ◽

Vol 2 (3) ◽

pp. 195-226 ◽

Cited By ~ 12

Author(s):

F. Pasquarelli ◽

A. Quarteroni ◽

G. Sacchi-Landriani

Keyword(s):

Stokes Problem ◽

Spectral Approximations ◽

Divergence Free

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Piecewise Divergence-Free Nonconforming Virtual Elements for Stokes Problem in Any Dimensions

SIAM Journal on Numerical Analysis ◽

10.1137/20m1350479 ◽

2021 ◽

Vol 59 (3) ◽

pp. 1835-1856

Author(s):

Huayi Wei ◽

Xuehai Huang ◽

Ao Li

Keyword(s):

Stokes Problem ◽

Divergence Free

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Guaranteed Upper Bounds For The Velocity Error Of Pressure-Robust Stokes Discretisations

Journal of Numerical Mathematics ◽

10.1515/jnma-2021-0078 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

P.L. Lederer ◽

C. Merdon

Keyword(s):

Mixed Methods ◽

Error Control ◽

Stokes Problem ◽

Upper Bounds ◽

Type Result ◽

Numerical Examples ◽

Velocity Error ◽

Divergence Free ◽

Pressure Independent ◽

Primal And Dual

Abstract This paper aims to improve guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the velocity errors of divergence-free primal and perfectly equilibrated dual mixed methods for the velocity stress. The first main result of the paper is a framework with relaxed constraints on the primal and dual method. This enables to use a recently developed mass conserving mixed stress discretisation for the design of equilibrated fluxes and to obtain pressure-independent guaranteed upper bounds for any pressure-robust (not necessarily divergence-free) primal discretisation. The second main result is a provably efficient local design of the equilibrated fluxes with comparably low numerical costs. Numerical examples verify the theoretical findings and show that efficiency indices of our novel guaranteed upper bounds are close to one.

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