Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-04T21:12:00.754Z Has data issue: false hasContentIssue false
  • English
  • Français

An Artificial Boundary Condition for a Class of Quasi-Newtonian Stokes Flows

Published online by Cambridge University Press:  28 May 2015

Baoqing Liu ,
Qing Chen  and
Qikui Du
Baoqing Liu*
Affiliation:
School of Applied Mathematics, Nanjing University of Finance and Economics, Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing 210023, China
Qing Chen
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing 210023, China
Qikui Du*
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing 210023, China
*
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
Get access

Abstract

An artificial boundary condition method, derived in terms of infinite Fourier series, is applied to solve a class of quasi-Newtonian Stokes flows. Based on the natural boundary reduction involving an artificial condition on the artificial boundary, the coupled variational problem and its numerical solution are obtained. The unique solvability of the continuous and discrete formulations are discussed, and the error analysis for the problem is also considered. Finally, an a posteriori error estimate for the corresponding problem is provided.

Keywords

65H0565B99Stokes equationartificial boundary conditionfinite element method
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 1 , February 2014 , pp. 35 - 51
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Baranger, J., Najib, K., and Sandri, D., Numerical analysis of a three-fields model for a quasi-Newtonian flow, Comput. Methods Appl. Mech. Engrg. 109, 282–292 (1993).Google Scholar
[2]Gatica, G. and Hsiao, G., Boundary-Field Equation Methods for a Class of Nonlinear Problems, Pitman Research Notes in Mathematics Series, vol. 331, Longman, Harlow, UK, 1995.Google Scholar
[3]Araya, R., Gatica, G., and Mennickent, H., A Boundary-field equation method for a nonlinear exterior elasticity problem in the plane, J. Elasticity 43, 45–68 (1996).Google Scholar
[4]Carstensen, C. and Gwinner, J., FEM and BEM coupling for a nonlinear transmission problem with Signorini contact, SIAM J. Numer. Anal. 34, 1845–1864 (1997).Google Scholar
[5]Feng, K., Finite element method and natural boundary reduction, Proceedings of the International Congress of Mathematicians, Ciesielski, Z., Olech, C. (ed.), Warszawa, vol. 1, pp. 1439–1453, 1983.Google Scholar
[6]Grote, M. and Keller, J., On non-reflecting boundary conditions, J. Comput. Phys. 122, 231–243 (1995).Google Scholar
[7]Gatica, G. and Hsiao, G., On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in ℝ2, Numer. Math. 61, 171–214 (1992).CrossRefGoogle Scholar
[8]Han, H. and Bao, W., The artificial boundary conditions for incompressible materials on an unbounded domain, Numer. Math. 77, 347–363 (1997).Google Scholar
[9]Han, H. and Wu, X., The Artificial Boundary Method-Numerical Solutions of Partial Differential Equations on Unbounded Domains (in Chinese), Tsinghua University Press, 2010.Google Scholar
[10]Costabel, M. and Stephan, E., Coupling of finite and boundary element methods for an elasto-plastic interface problem, SIAM J. Numer. Anal. 27, 1212–1226 (1990).CrossRefGoogle Scholar
[11]Feistauer, M., Mathematical and numerical study of nonlinear problems influid mechanics, Proceedings of the International Conference on Differential Equations and their Applications, Vosmanshy, J., Zlámal, M. (ed.), Springer, pp. 3–16, 1986.Google Scholar
[12]Gatica, G., Gatica, L., and Stephan, E., A FEM-DtN formulation for a non-linear exterior problem in incompressible elasticity, Math. Methods Appl. Sci. 26, 151–170 (2003).Google Scholar
[13]Keller, J. and Grote, M., Exact non-reflecting boundary conditions, J. Comput. Phys. 82, 172–192 (1989).CrossRefGoogle Scholar
[14]Gatica, G. and Stephan, E., A mixed-FEM formulation for nonlinear in compressible elasticity in the plane, Numer. Methods Partial Differential Equtaions 18, 105–128 (2002).Google Scholar
[15]Brink, U. and Stephan, E., Adaptive coupling of boundary elements and mixed finite elements for incompressible elasticity, Numer. Methods Partial Differential Equtaions 17, 79–92 (2001).3.0.CO;2-O>CrossRefGoogle Scholar
[16]Gatica, G., González, M., and Meddahi, S., A lower-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Parti: apriori error analysis, Comput. Methods Appl. Mech. Engrg. 193, 881–892 (2004).Google Scholar
[17]Gatica, G., González, M., and Meddahi, S., A lower-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part II: a posterior error analysis, Comput. Methods Appl. Mech. Engrg. 193, 893–911 (2004).CrossRefGoogle Scholar
[18]Loula, A. and Guerreior, J., Finite element analysis of nonlinear creeping flows, Comput. Methods Appl. Mech. Engrg. 79, 87–109 (1990).Google Scholar
[19]Yu, D. and Huang, H., The artificial boundary method for a nonlinear interface problem on unbounded domain, Comput. Model. Engrg. Sci. 35, 227–252 (2008).Google Scholar
[20]Yu, D., Natural Boundary Integral Method and Its Applications, Science Press & Kluwer Academic Publishers, 2002.Google Scholar
[21]Zheng, C. and Han, H., Artificial boundary method for the exterior Stokes flow in three dimensions, Int. J. Numer. Methods Fluids 41, 537–549 (2003).Google Scholar
[22]Scheurer, B., Existence et approximation de point selles pour certain problems non-linéaires, RAIRO Anal. Numer. 11, 369–400 (1977).CrossRefGoogle Scholar
[23]Girault, V. and Raviart, P., Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag, 1986.Google Scholar