A nonconforming finite element method for the stationary Smagorinsky model

Abstract In this paper, we focus on a low order nonconforming finite element method (FEM) for the stationary Smagorinsky model. The velocity and pressure are approximated by the constrained nonconforming rotated Q1 element (CN Q 1 r o t ) and piecewise constant element, respectively. Optimal error estimates of the velocity in the broken H1-norm and L2-norm, and the pressure in the L2-norm are derived by some nonlinear analysis techniques and Aubin-Nitsche duality argument. The supercloseness and superconvergent results are also obtained under some reasonable regularity assumptions. Finally, a numerical example is implemented to confirm our theoretical analysis.

[1]  Qiang Du,et al.  Finite-element approximations of a Ladyzhenskaya model for stationary incompressible viscous flow , 1990 .

[2]  Jincheng Ren,et al.  A new second order nonconforming mixed finite element scheme for the stationary Stokes and Navier-Stokes equations , 2009, Appl. Math. Comput..

[3]  J. Douglas,et al.  A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier–Stokes equations , 1999 .

[4]  Volker John,et al.  Finite Element Methods for Incompressible Flow Problems , 2016 .

[5]  Xinlong Feng,et al.  Three Iterative Finite Element Methods for the Stationary Smagorinsky Model , 2014 .

[6]  Zhangxin Chen,et al.  A new local stabilized nonconforming finite element method for solving stationary Navier-Stokes equations , 2011, J. Comput. Appl. Math..

[7]  Shao-chunChen,et al.  AN ANISOTROPIC NONCONFORMING FINITE ELEMENT WITH SOME SUPERCONVERGENCE RESULTS , 2005 .

[8]  Dongwoo Sheen,et al.  A cheapest nonconforming rectangular finite element for the stationary Stokes problem , 2013 .

[9]  Yu’e Ma,et al.  A superconvergent nonconforming mixed finite element method for the Navier–Stokes equations , 2016 .

[10]  Chunmei Xie,et al.  New nonconforming finite element method for solving transient Naiver-Stokes equations , 2014 .

[11]  Shi,et al.  CONSTRAINED QUADRILATERAL NONCONFORMING ROTATED Q1 ELEMENT , 2005 .

[12]  R. Rannacher,et al.  Simple nonconforming quadrilateral Stokes element , 1992 .

[13]  Yuan Li,et al.  Error estimates of two-level finite element method for Smagorinsky model , 2016, Appl. Math. Comput..

[14]  Traian Iliescu,et al.  A Two-Level Discretization Method for the Smagorinsky Model , 2008, Multiscale Model. Simul..

[15]  Dongwoo Sheen,et al.  P1-Nonconforming Quadrilateral Finite Element Methods for Second-Order Elliptic Problems , 2003, SIAM J. Numer. Anal..

[16]  J. Smagorinsky,et al.  GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS , 1963 .

[17]  Shipeng Mao,et al.  A quadrilateral, anisotropic, superconvergent, nonconforming double set parameter element , 2006 .

[18]  Q. Lin,et al.  Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation , 2005 .

[19]  Xinlong Feng,et al.  Two-level stabilized method based on Newton iteration for the steady Smagorinsky model☆ , 2013 .

[20]  Ningning Yan,et al.  Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations , 2008, Adv. Comput. Math..

[21]  Dongwoo Sheen,et al.  Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems , 1999 .

[22]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[23]  R. D. Richtmyer,et al.  A Method for the Numerical Calculation of Hydrodynamic Shocks , 1950 .

[24]  D. Shi,et al.  The Crouzeix-Raviart type nonconforming finite element method for the nonstationary Navier-Stokes equations on anisotropic meshes , 2014 .

[25]  Jincheng Ren,et al.  Nonconforming mixed finite element approximation to the stationary Navier–Stokes equations on anisotropic meshes , 2009 .

[26]  Dongyang Shi,et al.  Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations , 2016 .