Analysis of Compatible Discrete Operator Schemes for the Stokes Equations on Polyhedral Meshes

Jérôme Bonelle (1) , Alexandre Ern (2)

1 EDF - EDF
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

Compatible Discrete Operator schemes preserve basic properties of the continuous model at the discrete level. They combine discrete differential operators that discretize exactly topological laws and discrete Hodge operators that approximate constitutive relations. We devise and analyze two families of such schemes for the Stokes equations in curl formulation, with the pressure degrees of freedom located at either mesh vertices or cells. The schemes ensure local mass and momentum conservation. We prove discrete stability by establishing novel discrete Poincaré inequalities. Using commutators related to the consistency error, we derive error estimates with first-order convergence rates for smooth solutions. We analyze two strategies for discretizing the external load, so as to deliver tight error estimates when the external load has a large irrotational or divergence-free part. Finally, numerical results are presented on three-dimensional polyhedral meshes.

Mots clés: compatible discretization; mimetic discretization; Stokes flows; polyhedral meshes; CDO schemes; error bounds

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Jérôme Bonelle, Alexandre Ern. Analysis of Compatible Discrete Operator Schemes for the Stokes Equations on Polyhedral Meshes. 2014. ⟨hal-00939164v1⟩