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

Antti Hannukainen (1) , Rolf Stenberg (1) , Martin Vohralík (2)

1 ONERA - The French Aerospace Lab [Châtillon]
2 LJLL - Laboratoire Jacques-Louis Lions

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.

Keywords: Stokes problem; A posteriori error estimate; Guaranteed upper bound; Unified framework; Conforming finite element method; Discontinuous Galerkin method; Nonconforming finite element method; Mixed finite element method; Finite volume method

Domains: Numerical Analysis [math.NA]

Antti Hannukainen, Rolf Stenberg, Martin Vohralík. A unified framework for a posteriori error estimation for the Stokes problem. Numerische Mathematik, 2012, 122 (4), pp.725-769. ⟨10.1007/s00211-012-0472-x⟩. ⟨hal-00470131v2⟩