A multilevel finite element method in space‐time for the Navier‐Stokes problem

A multilevel finite element method in space-time for the two-dimensional nonstationary Navier-Stokes problem is considered. The method is a multi-scale method in which the fully nonlinear Navier-Stokes problem is only solved on a single coarsest space-time mesh; subsequent approximations are generated on a succession of refined space-time meshes by solving a linearized Navier-Stokes problem about the solution on the previous level. The a priori estimates and error analysis are also presented for the J-level finite element method. We demonstrate theoretically that for an appropriate choice of space and time mesh widths: hj ∼ h, kj ∼ k, j = 2, …, J, the J-level finite element method in space-time provides the same accuracy as the one-level method in space-time in which the fully nonlinear Navier-Stokes problem is solved on a final finest space-time mesh. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

[1]  Jie Shen.  Long time stability and convergence for fully discrete nonlinear galerkin methods , 1990 .

[2]  Yinnian He,et al.  Two-Level Method Based on Finite Element and Crank-Nicolson Extrapolation for the Time-Dependent Navier-Stokes Equations , 2003, SIAM J. Numer. Anal..

[3]  H. K. Lee,et al.  Numerical Solution of the Stationary Navier-Stokes Equations Using a Multilevel Finite Element Method , 1998, SIAM J. Sci. Comput..

[4]  Weiwei Sun,et al.  Multi-level spectral galerkin method for the navier-stokes problem I : spatial discretization , 2005, Numerische Mathematik.

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

[6]  R. Rannacher,et al.  Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization , 1990 .

[7]  R. Kellogg,et al.  A regularity result for the Stokes problem in a convex polygon , 1976 .

[8]  Maxim A. Olshanskii,et al.  Two-level method and some a priori estimates in unsteady Navier—Stokes calculations , 1999 .

[9]  William Layton,et al.  A multilevel mesh independence principle for the Navier-Stokes equations , 1996 .

[10]  Yinnian He,et al.  Multi-level spectral Galerkin method for the Navier–Stokes equations, II: time discretization , 2006, Adv. Comput. Math..

[11]  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 .

[12]  Vivette Girault,et al.  Two-grid finite-element schemes for the transient Navier-Stokes problem , 2001 .

[13]  Jinchao Xu,et al.  A Novel Two-Grid Method for Semilinear Elliptic Equations , 1994, SIAM J. Sci. Comput..

[14]  Jinchao Xu.  Two-grid Discretization Techniques for Linear and Nonlinear PDEs , 1996 .

[15]  Endre Süli,et al.  Approximation of the global attractor for the incompressible Navier–Stokes equations , 2000 .

[16]  Lutz Tobiska,et al.  A Two-Level Method with Backtracking for the Navier--Stokes Equations , 1998 .

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

[18]  William Layton,et al.  A two-level discretization method for the Navier-Stokes equations , 1993 .