Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems

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 (T. Boiveau & E. Burman. IMA J. Numer. Anal. 36 (2016), no.2, 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 flows, 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.