A projection method based on self-adaptive rules for Stokes equations with nonlinear slip boundary conditions. (English) Zbl 07244666
Summary: In this work we propose and study a self-adaptive projection method for the numerical solution of Stokes equations under slip boundary conditions. We introduce the projection operator to formulate the slip constraint as a fixed point equation. To solve this problem we suggest a projection method by reformulating the nonlinear slip boundary conditions into an iterative form. To make the method more efficient, we find a self-adaptive rule that uses iterative functions to adjust the penalty parameter automatically. We show the convergence of the method in function space and give its application in detail. Finally, the numerical results are given to support our theoretical results.
References:
| [1] | Ayadi, M.; Baffico, L.; Gdoura, M. K.; Sassi, T., Error estimates for Stokes problem with Tresca friction conditions, ESAIM: Math. Model. Numer. Anal., 48, 1413-1429 (2014) · Zbl 1381.76153 |
| [2] | Bnouhachem, A., A modified inexact operator splitting method for monotone variational inequalities, J. Glob. Optim., 41, 417-426 (2008) · Zbl 1158.65045 |
| [3] | Chouly, F., An adaptation of Nitsche’s method to the Tresca friction problem, J. Math. Anal. Appl., 411, 329-339 (2014) · Zbl 1311.74112 |
| [4] | Djoko, J.; Koko, J., Numerical methods for the Stokes and Navier-Stokes equations driven by threshold slip boundary conditions, Comput. Methods Appl. Mech. Eng., 305, 936-958 (2016) · Zbl 1423.76224 |
| [5] | Fujita, H., A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, RIMS Kokyuroku, 88, 199-216 (1994) · Zbl 0939.76527 |
| [6] | Fujita, H.; Kawarada, H., Variational inequalities for the Stokes equation with boundary conditions of friction type, Int. Ser. Math. Sci. Appl., 11, 15-33 (1998) · Zbl 0926.76092 |
| [7] | Glowinski, R., Numerical Methods for Nonlinear Variational Problems (2008), Springer-Verlag: Springer-Verlag Berlin · Zbl 1139.65050 |
| [8] | Glowinski, R.; Le Tallec, P., Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, Studies in Applied and Numerical Mathematics (1989), SIAM: SIAM Philadelphia · Zbl 0698.73001 |
| [9] | Han, D. R., Solving linear variational inequality problems by a self-adaptive projection method, Appl. Math. Comput., 182, 1765-1771 (2006) · Zbl 1110.65057 |
| [10] | He, B. S., Self-adaptive operator splitting methods for monotone variational inequalities, Numer. Math., 94, 715-737 (2003) · Zbl 1033.65047 |
| [11] | Jing, F. F.; Han, W. M.; Yan, W. J.; Wang, F., Discontinuous Galerkin methods for a stationary Navier-Stokes problem with a nonlinear slip boundary conditions of friction type, J. Sci. Comput., 76, 888-912 (2018) · Zbl 1397.65272 |
| [12] | Kashiwabara, T., On a finite element approximation of the Stokes equations under a slip boundary conditions of the friction type, Jpn. J. Ind. Appl. Math., 30, 227-261 (2013) · Zbl 1263.76043 |
| [13] | Le Roux, C., Steady Stokes flows with threshold slip boundary conditions, Math. Models Methods Appl. Sci., 15, 1141-1168 (2005) · Zbl 1129.35424 |
| [14] | Li, Y.; Li, K. T., Existence of the solution to stationary Navier-Stokes equations with nonlinear slip boundary conditions, J. Math. Anal. Appl., 381, 1-9 (2011) · Zbl 1221.35282 |
| [15] | Li, Y.; Li, K. T., Uzawa iteration method for Stokes type variational inequality of the second kind, Acta Math. Appl. Sin. Engl. Ser., 27, 303-316 (2011) · Zbl 1211.35220 |
| [16] | Noor, M. A., Some developments in general variational inequalities, Appl. Math. Comput., 152, 199-277 (2004) · Zbl 1134.49304 |
| [17] | Saito, N., On the Stokes equations with the leak and slip boundary conditions of friction type: regularity of solutions, Publ. Res. Inst. Math. Sci. Kyoto Univ., 40, 345-383 (2004) · Zbl 1050.35029 |
| [18] | Zhang, S. G., Projection and self-adaptive projection methods for the Signorini problem with the BEM, Comput. Math. Appl., 74, 1262-1273 (2017) · Zbl 1398.65327 |
| [19] | Zhang, S. G., Two projection methods for the solution of Signorini problems, Appl. Math. Comput., 326, 75-86 (2018) · Zbl 1426.65195 |
| [20] | Zhang, S. G.; Li, X. L., A self-adaptive projection method for contact problems with the BEM, Appl. Math. Model., 55, 145-159 (2018) · Zbl 1480.74235 |
| [21] | Zhang, S. G.; Li, X. L.; Ran, R. S., Self-adaptive projection and boundary element methods for contact problems with Tresca friction, Commun. Nonlinear Sci. Numer. Simul., 68, 72-85 (2019) · Zbl 1508.74067 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
