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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.
Blank, Laura 1 ; Caiazzo, Alfonso 1 ; Chouly, Franz 1 ; Lozinski, Alexei 1 ; Mura, Joaquin 1
@article{M2AN_2018__52_6_2149_0,
author = {Blank, Laura and Caiazzo, Alfonso and Chouly, Franz and Lozinski, Alexei and Mura, Joaquin},
title = {Analysis of a stabilized penalty-free {Nitsche} method for the {Brinkman,} {Stokes,} and {Darcy} problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2149--2185},
publisher = {EDP-Sciences},
volume = {52},
number = {6},
year = {2018},
doi = {10.1051/m2an/2018063},
zbl = {1417.65196},
mrnumber = {3905192},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018063/}
}
TY - JOUR AU - Blank, Laura AU - Caiazzo, Alfonso AU - Chouly, Franz AU - Lozinski, Alexei AU - Mura, Joaquin TI - Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 2149 EP - 2185 VL - 52 IS - 6 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018063/ DO - 10.1051/m2an/2018063 LA - en ID - M2AN_2018__52_6_2149_0 ER -
%0 Journal Article %A Blank, Laura %A Caiazzo, Alfonso %A Chouly, Franz %A Lozinski, Alexei %A Mura, Joaquin %T Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 2149-2185 %V 52 %N 6 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018063/ %R 10.1051/m2an/2018063 %G en %F M2AN_2018__52_6_2149_0
Blank, Laura; Caiazzo, Alfonso; Chouly, Franz; Lozinski, Alexei; Mura, Joaquin. Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2149-2185. doi: 10.1051/m2an/2018063
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