An a posteriori error estimation for the discrete duality finite volume discretization of the Stokes equations

¹ CEA, DEN, DM2S-STMF, 91191 Gif-sur-Yvette cedex, France.
anhha.le84@gmail.com; pascal.omnes@cea.fr
² Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS, UMR 7539, 99, Avenue J.-B. Clément 93430 Villetaneuse cedex, France.

Received: 11 July 2012
Revised: 2 September 2014

Abstract

We derive an a posteriori error estimation for the discrete duality finite volume (DDFV) discretization of the stationary Stokes equations on very general twodimensional meshes, when a penalty term is added in the incompressibility equation to stabilize the variational formulation. Two different estimators are provided: one for the error on the velocity and one for the error on the pressure. They both include a contribution related to the error due to the stabilization of the scheme, and a contribution due to the discretization itself. The estimators are globally upper as well as locally lower bounds for the errors of the DDFV discretization. They are fully computable as soon as a lower bound for the inf-sup constant is available. Numerical experiments illustrate the theoretical results and we especially consider the influence of the penalty parameter on the error for a fixed mesh and also of the mesh size for a fixed value of the penalty parameter. A global error reducing strategy that mixes the decrease of the penalty parameter and adaptive mesh refinement is described.