A unified framework for a posteriori error estimation for the Stokes problem

Abstract : In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on $[H^1_0(\Om)]^d$-conforming velocity reconstruction and $\HH(\dv,\Om)$-conforming, locally conservative flux (stress) reconstruction. It gives guaranteed, fully computable global upper bounds as well as local lower bounds on the energy error. In order to apply this framework to a given numerical method, two simple conditions need to be checked. We show how to do this for various conforming and conforming stabilized finite element methods, the discontinuous Galerkin method, the Crouzeix--Raviart nonconforming finite element method, the mixed finite element method, and a general class of finite volume methods. Numerical experiments illustrate the theoretical developments.
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Antti Hannukainen, Rolf Stenberg, Martin Vohralík. A unified framework for a posteriori error estimation for the Stokes problem. Numerische Mathematik, Springer Verlag, 2012, 122 (4), pp.725-769. ⟨10.1007/s00211-012-0472-x⟩. ⟨hal-00470131v2⟩

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