Volume 52, Number 6, November-December 2018
|Page(s)||2149 - 2185|
|Published online||01 February 2019|
Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, 10117 Berlin, Germany
2 Institut de Mathématiques de Bourgogne, UMR CNRS 5584, Université Bourgogne Franche-Comté, 9 avenue Alain Savary BP 47870, 21078 Dijon Cedex, France
3 Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université Bourgogne Franche Comté, 16 route de Gray, 25030 Besançon Cedex, France
4 Biomedical Imaging Center, School of Engineering and School of Medicine, Pontificia Universidad Católica de Chile Av. Vicuña Mackenna 4860, 7820436 Santiago, Chile
Accepted: 25 October 2018
In this paper we study the Brinkman model as a unified framework to allow the transition between the Darcy and the Stokes problems. We propose an unconditionally stable low-order finite element approach, which is robust with respect to the whole range of physical parameters, and is based on the combination of stabilized equal-order finite elements with a non-symmetric penalty-free Nitsche method for the weak imposition of essential boundary conditions. In particular, we study the properties of the penalty-free Nitsche formulation for the Brinkman setting, extending a recently reported analysis for the case of incompressible elasticity (Boiveau and Burman, IMA J. Numer. Anal. 36 (2016) 770-795). Focusing on the two-dimensional case, we obtain optimal a priori error estimates in a mesh-dependent norm, which, converging to natural norms in the cases of Stokes or Darcy ows, allows to extend the results also to these limits. Moreover, we show that, in order to obtain robust estimates also in the Darcy limit, the formulation shall be equipped with a Grad-Div stabilization and an additional stabilization to control the discontinuities of the normal velocity along the boundary. The conclusions of the analysis are supported by numerical simulations.
Mathematics Subject Classification: 65N30 / 65N12 / 65N15
Key words: Brinkman problem / penalty-free Nitsche method / weak boundary conditions / stabilized finite elements
© EDP Sciences, SMAI 2019
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