The Stokes flow with the stick-slip boundary condition combining the Navier and Tresca laws is considered. The finite element approximation leads to an algebraic optimization problem. Its optimality condition is the starting point for an active set implementation of the semi-smooth Newton method. Numerical experiments demonstrate computational efficiency of an adaptive diagonal preconditioner.

Topics
Newton Raphson method, Optimization problems, Laminar flows
1.
M.
 
Ayadi
,
L.
 
Baffico
,
M. K.
 
Gdoura
,
T.
 
Sassi
,
Error estimates for Stokes problem with Tresca friction conditions
,
ESAIM: Mathematical Modelling and Numerical Analysis
,
48
,
1413
–
1429
(
2014
).
https://doi.org/10.1051/m2an/2014001
Google Scholar
Crossref
Search ADS
 
2.
J. K.
 
Djoko
,
M.
 
Mbehou
,
Finite element analysis for Stokes and Navier Stokes equations driven by threshold slip boundary conditions
,
Int. J. Numer. Anal. Model.
, Series B,
4
,
235
–
255
(
2013
).
Google Scholar
 
3.
H.
 
Fujita
,
A mathematical analysis of motions of viscous incompressible fluid under leak and slip boundary conditions
,
RIMS Kokyuroku
,
888
,
199
–
216
(
1994
).
Google Scholar
 
4.
J.
 
Haslinger
,
R.
 
Kučera
, and
V.
 
Šátek
,
Stokes flow with slip boundary conditions of Coulomb type
, accepted in
Mathematics and Mechanics of Solids
(
2017
).
Google Scholar
 
5.
M.
 
Hintermüller
,
K.
 
Ito
,
K.
 
Kunisch
,
The primal-dual active set strategy as a semismooth Newton method
,
SIAM J. Optim.
,
13
,
865
–
888
(
2003
).
https://doi.org/10.1137/S1052623401383558
Google Scholar
Crossref
Search ADS
 
6.
R.
 
Kučera
,
J.
 
Machalová
,
H.
 
Netuka
,
P.
 
Ženčák
,
An interior point algorithm for the minimization arising from 3D contact problems with friction
,
Optimization Methods and Software
 
28
,
1195
–
1217
(
2013
).
https://doi.org/10.1080/10556788.2012.684352
Google Scholar
Crossref
Search ADS
 
7.
R.
 
Kučera
,
J.
 
Haslinger
,
V.
 
Šátek
,
M.
 
Jarošová
,
Efficient methods for solving the Stokes problem with slip boundary conditions
, accepted in
Mathematics and Computers in Simulation
(
2017
),
https://doi.org/10.1016/j.matcom.2016.05.012
.
Google Scholar
 
8.
J.
 
Koko
,
Vectorized Matlab codes for the Stokes problem with P1-bubble/P1 finite element
, at: http://www.isima.fr/∼jkoko/Codes/StokesP1BubbleP1.pdf.
9.
J.
 
Pacholek
,
Semi-smooth Newton method for solving the Stokes equations with monotonously increasing slip condition
, Diploma thesis,
VŠB-TU Ostrava
(
2017
).
10.
I.
 
Rao
,
K.
 
Rajagopal
,
The effect of the slip boundary conditions on the flow of fluids in a channel
,
Acta Mechanica
 
135
,
113
–
126
(
1999
).
https://doi.org/10.1007/BF01305747
Google Scholar
Crossref
Search ADS
 
This content is only available via PDF.
© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.