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

• Published in: Numerische Mathematik Vol. 122; no. 4; pp. 725 - 769
• Main Authors: Hannukainen, Antti, Stenberg, Rolf, Vohralík, Martin
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.12.2012; Springer Verlag
• Subjects: Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Simulation; Theoretical; 65N15; 76S05; 76M12; Discontinuous Galerkin method; Mixed finite element method; Stokes problem; A posteriori error estimate; Conforming finite element method; Unified framework; Guaranteed upper bound; Finite volume method; Nonconforming finite element method
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-012-0472-x

In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on -conforming velocity reconstruction and -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. The tools developed and used include a new simple equilibration on dual meshes and the solution of local Poisson-type Neumann problems by the mixed finite element method. Numerical experiments illustrate the theoretical developments.