Superconvergence in velocity and pressure for the 3D time-dependent Navier-Stokes Equations

This work is devoted to the superconvergence in space approximation of a fully discrete scheme for the incompressible time-dependent Navier-Stokes Equations in three-dimensional domains. We discrete by Inf-Sup-stable Finite Element in space and by a semi-implicit backward Euler (linear) scheme in time.Using an extension of the duality argument in negative-norm for elliptic linear problems (see for instance [1]) to the mixed velocity-pressure formulation of the Stokes problem, we prove some superconvergence in space results for the velocity with respect to the energy-norm, and for a weaker norm of L2(0, T; L2(Ω)) type (this latter holds only for the case of Taylor-Hood approximation). On the other hand, we also obtain optimal error estimates for the pressure without imposing constraints on the time and spatial discrete parameters, arriving at superconvergence in the H1 (Ω)-norm again for Taylor-Hood approximations. These results are numerically verified by several computational experiments, where two splitting in time schemes are also considered.

[1]  Weiwei Sun,et al.  Stability and Convergence of the Crank-Nicolson/Adams-Bashforth scheme for the Time-Dependent Navier-Stokes Equations , 2007, SIAM J. Numer. Anal..

[2]  R. Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization , 1982 .

[3]  Yinnian He,et al.  The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data , 2008, Math. Comput..

[4]  M. Wheeler.  A Priori L_2 Error Estimates for Galerkin Approximations to Parabolic Partial Differential Equations , 1973 .

[5]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[6]  Ricardo H. Nochetto,et al.  Stability of the finite element Stokes projection in W1 , 2004 .

[7]  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 .

[8]  F. Thomasset.  Finite element methods for Navier-Stokes equations , 1980 .

[9]  Rolf Stenberg,et al.  A technique for analysing finite element methods for viscous incompressible flow , 1990 .

[10]  V. Girault,et al.  Estimates of the finite element Stokes projection in W 1 , ∞ Stabilité dans W 1 , ∞ de la projection de Stokes par éléments finis , 2004 .

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

[12]  Junping Wang,et al.  Superconvergence of Finite Element Approximations for the Stokes Problem by Projection Methods , 2001, SIAM J. Numer. Anal..

[13]  Mary Fanett.  A PRIORI L2 ERROR ESTIMATES FOR GALERKIN APPROXIMATIONS TO PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS , 1973 .

[14]  A. Prohl.  Projection and quasi-compressibility methods for solving the incompressible navier-stokes equations , 1997 .

[15]  R. Temam,et al.  Navier-stokes equations: Theory and approximation , 1998 .

[16]  R. Temam.  Navier-Stokes Equations , 1977 .

[17]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[18]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

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