A Posteriori Error Estimates for Finite Element Approximation of Unsteady Incompressible Stochastic Navier-Stokes Equations

In this paper, we consider the unsteady incompressible stochastic Navier-Stokes equations containing a noise part. The purpose of this paper is to derive a posteriori error estimates for the finite element approximation of stochastic equations applied the weighted Clement-type interpolator. We obtain the posteriori error upper and lower bounds for the semidicretization scheme and the full backward Euler discretization scheme.

[1]  Carsten Carstensen,et al.  A unifying theory of a posteriori finite element error control , 2005, Numerische Mathematik.

[2]  Alain Bensoussan,et al.  Stochastic Navier-Stokes Equations , 1995 .

[3]  Yubin Yan,et al.  Galerkin Finite Element Methods for Stochastic Parabolic Partial Differential Equations , 2005, SIAM J. Numer. Anal..

[4]  Jacques Simon,et al.  On the Existence of the Pressure for Solutions of the Variational Navier—Stokes Equations , 1999 .

[5]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[6]  Ruo Li,et al.  A Posteriori Error Estimates of Recovery Type for Distributed Convex Optimal Control Problems , 2007, J. Sci. Comput..

[7]  Jonathan C. Mattingly.  On recent progress for the stochastic Navier Stokes equations , 2004, math/0409194.

[8]  Christine Bernardi,et al.  Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem , 2006, Numerische Mathematik.

[9]  W. B. Liu,et al.  Quasi-norm Error Bounds for the Nite Element Approximation of a Non-newtonian Ow , 1994 .

[10]  Kwang Y. Kim.  A posteriori error estimators for locally conservative methods of nonlinear elliptic problems , 2007 .

[11]  Wenbin Liu,et al.  A Posteriori Error Estimates for Finite Element Approximation of Parabolic p-Laplacian , 2006, SIAM J. Numer. Anal..

[12]  Wenbin Liu,et al.  Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian , 2001, Numerische Mathematik.

[13]  Yinnian He,et al.  Two-level Stabilized Finite Element Methods for the Steady Navier–Stokes Problem , 2005, Computing.

[14]  W. B. Liu,et al.  Quasi-Norm interpolation error estimates for the piecewise linear finite element approximation of p-Laplacian problems , 2005, Numerische Mathematik.

[15]  Ningning Yan,et al.  A Posteriori Error Estimators of Gradient Recovery Type for Elliptic Obstacle Problems , 2001, Adv. Comput. Math..

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

[17]  José A. Langa,et al.  Existence and Regularity of the Pressure for the Stochastic Navier–Stokes Equations , 2003 .

[18]  C. Carstensen.  QUASI-INTERPOLATION AND A POSTERIORI ERROR ANALYSIS IN FINITE ELEMENT METHODS , 1999 .

[19]  Wei Wang,et al.  The approximation of a Crank-Nicolson scheme for the stochastic Navier-Stokes equations , 2009 .

[20]  M. Capinski,et al.  On the existence of a solution to stochastic Navier—Stokes equations , 2001 .

[21]  Wenbin Liu,et al.  Quasi-Norm Local Error Estimators for p-Laplacian , 2001, SIAM J. Numer. Anal..