Optimal Error Estimate of the Penalty Finite Element Method for the Micropolar Fluid Equations

REFERENCES

• M. Bercovier ( 1978 ). Perturbation of mixed variational problem, applications to mixed finite element methods . RAIRO Anal. Numer. 12 : 221 – 236 .
   Google Scholar
• B. Brefort ( 1998 ). Attractor for the penalty for the Navier-Stokes equations . SIAM J. Math. Anal. 19 : 1 – 21 .
   Google Scholar
• P.G. Ciarlet ( 1978 ). The Finite Element Method for Elliptic Problems . North-Holland , Amsterdam .
   Google Scholar
• D.W. Condiff and J.S. Dahler ( 1964 ). Fluid mechanics aspects of antisymmetric stress . Phys. Fluids 11 : 842 – 854 .
   Web of Science ®Google Scholar
• A.C. Eringen ( 1966 ). Theory of micropolar fluids . J. Math. Mech. 16 : 1 – 8 .
   Web of Science ®Google Scholar
• V. Girault and P.A. Raviart ( 1987 ). Finite Elements Methods for Navier-Stokes Equation: Theory and Algorithms . Springer-Verlarg , Berlin , Heidelberg .
   Google Scholar
• Yinnian He ( 2005 ). Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes equations . Math. Computat 74 , 251 : 1201 – 1216 .
   Google Scholar
• J.G. Heywood and R. Rannacher ( 1990 ). Finite element approximation of the nonstationary Navier-Stokes problem IV: Error analysis for second-order time discretization . SIAM J. Numer. Anal. 27 : 353 – 384 .
   Web of Science ®Google Scholar
• J.R. Hughes , W.T. Liu , and A.J. Brooks ( 1979 ). Finite element analysis of incompressible viscous flows by the penalty function formulation . J. Comp. Phys. 30 : 1 – 60 .
   Web of Science ®Google Scholar
• O.A. Ladyzhenskaya ( 1969 ). The Mathematical Theory of Viscous Incompressible Flow. , Second edition . Gordon and Breach , New York .
   Google Scholar
• G. Lukaszewicz ( 1989 ). On nonstationary flows of asymmetric fluids . Rend. Accad. Naz. Sovi. XL, Mem, Math. XII : 83 – 97 .
   Google Scholar
• G. Lukaszewicz ( 1989 ). On the existence, uniqueness and asymptotic properties of solutions of flows of asymmetric fluids . Rend. Accad. Naz. Sovi. XL, Mem, Math. XII : 105 – 120 .
   Google Scholar
• G. Lukaszewicz ( 1998 ). Micropolar Fluids:Theory and Applications . Birkhäuser , Berlin .
   Google Scholar
• L.G. Petrosyan ( 1984 ). Some Problems of Mechanics of Fluids with Antisymmetric Stress Tensor . Erevan (in Russian) .
   Google Scholar
• J. Shen (1995). On error estimates of the penalty method with backtraking for the Navier-Stokes equations. SIAM J. Numer. Anal. 32:386–403.
   Web of Science ®Google Scholar
• J. Shen ( 1992a ). On error estimates of some higher order projection and penalty-projection methods for the Navier-Stokes equations . Numer. Math. 62 : 49 – 73 .
   Web of Science ®Google Scholar
• J. Shen ( 1992b ). On error estimates of projection methods for the Navier-Stokes equations . SIAM Numer. Anal. 29 : 57 – 77 .
   Web of Science ®Google Scholar
• R. Temam ( 1968 ). Une méthode d'approximation de la solution des équations de Navier-Stokes . Bull. Soc. Math. France 98 : 115 – 152 .
   Google Scholar
• R. Temam ( 1979 ). Navier-Stokes Equations, Theory and Numerical Analysis. , 2nd rev. ed. North-Holland , Amsterdam .
   Google Scholar