Numerical Approximation of a Quasi-Newtonian Stokes Flow Problem with Defective Boundary Conditions

In this article we study the numerical approximation of a quasi-Newtonian Stokes flow problem where only the flow rates are specified at the inflow and outflow boundaries. A variational formulation of the problem, using Lagrange multipliers to enforce the stated flow rates, is given. The existence and the uniqueness to the continuous, and discrete, variational formulations of the solution are shown. An error analysis for the numerical approximation is also given. Numerical computations are included which demonstrate the approximation scheme studied.

[1]  Timothy Nigel Phillips,et al.  Residual a posteriori error estimator for a three-field model of a non-linear generalized Stokes problem , 2006 .

[2]  R. Rogers,et al.  An introduction to partial differential equations , 1993 .

[3]  R. Verfürth.  Finite element approximation on incompressible Navier-Stokes equations with slip boundary condition , 1987 .

[4]  G. Folland.  Introduction to Partial Differential Equations , 1976 .

[5]  Alfio Quarteroni,et al.  Numerical Treatment of Defective Boundary Conditions for the Navier-Stokes Equations , 2002, SIAM J. Numer. Anal..

[6]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[7]  M. Gunzburger,et al.  Treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses , 1992 .

[8]  G. Gatica,et al.  A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part I: a priori error analysis , 2004 .

[9]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[10]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[11]  C. Han,et al.  Multiphase flow in polymer processing , 1981 .

[12]  T N Phillips,et al.  Contemporary Topics in Computational Rheology , 2002 .

[13]  H. Manouzi,et al.  Mixed finite element analysis of a non‐linear three‐fields Stokes model , 2001 .

[14]  D. Sandri.  A posteriori estimators for mixed finite element approximations of a fluid obeying the power law , 1998 .

[15]  R. Bird.  Dynamics of Polymeric Liquids , 1977 .

[16]  Jacques Baranger,et al.  Numerical analysis of a three-fields model for a quasi-Newtonian flow , 1993 .

[17]  Rolf Rannacher,et al.  ARTIFICIAL BOUNDARIES AND FLUX AND PRESSURE CONDITIONS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS , 1996 .

[18]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .