Convergence analysis of a new multiscale finite element method for the stationary Navier-Stokes problem

In this paper, we propose a new multiscale finite element method for the stationary Navier-Stokes problem. This new method for the lowest equal order finite element pairs P"1/P"1 is based on the multiscale enrichment and derived from the Navier-Stokes problem itself. Therefore, the new multiscale finite element method better reflects the nature of the nonlinear problem. The well-posedness of this new discrete problem is proved under the standard assumption. Meanwhile, convergence of the optimal order in the H^1-norm for the velocity and the L^2-norm for the pressure is obtained. Especially, via applying a new dual problem and some techniques in the process for proof, we establish the convergence of the optimal order in the L^2-norm for the velocity. Finally, numerical examples confirm our theory analysis and validate the effectiveness of this new method.

[1]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[2]  David J. Silvester,et al.  ANALYSIS OF LOCALLY STABILIZED MIXED FINITE-ELEMENT METHODS FOR THE STOKES PROBLEM , 1992 .

[3]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[4]  P. Hansbo,et al.  Edge Stabilization for the Incompressible Navier-Stokes Equations: a Continuous Interior Penalty Finite Element Method , 2004 .

[5]  Ekkehard Ramm,et al.  A three-level finite element method for the instationary incompressible Navier?Stokes equations , 2004 .

[6]  Zhihao Ge,et al.  Analysis of multiscale finite element method for the stationary Navier–Stokes equations , 2012 .

[7]  Yinnian He,et al.  A stabilized finite element method based on local polynomial pressure projection for the stationary Navier--Stokes equations , 2008 .

[8]  Alessandro Russo,et al.  Bubble stabilization of finite element methods for the linearized incompressible Navier-Stokes equations , 1996 .

[9]  Frédéric Valentin,et al.  Convergence Analysis of a Residual Local Projection Finite Element Method for the Navier-Stokes Equations , 2012, SIAM J. Numer. Anal..

[10]  J. P. Benque,et al.  A finite element method for Navier-Stokes equations , 1980 .

[11]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of , 1986 .

[12]  Rodolfo Araya,et al.  A multiscale a posteriori error estimate , 2005 .

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

[14]  L. Franca,et al.  Error analysis of some Galerkin least squares methods for the elasticity equations , 1991 .

[15]  Franco Brezzi,et al.  Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S.) , 1993 .

[16]  Leopoldo P. Franca,et al.  On a two‐level finite element method for the incompressible Navier–Stokes equations , 2000 .

[17]  Frédéric Valentin,et al.  An unusual stabilized finite element method for a generalized Stokes problem , 2002, Numerische Mathematik.

[18]  P. Hansbo,et al.  A unified stabilized method for Stokes' and Darcy's equations , 2007 .

[19]  J. Douglas,et al.  Stabilized mixed methods for the Stokes problem , 1988 .

[20]  Thomas J. R. Hughes,et al.  The Stokes problem with various well-posed boundary conditions - Symmetric formulations that converge for all velocity/pressure spaces , 1987 .

[21]  Lutz Tobiska,et al.  Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations , 1996 .

[22]  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 .

[23]  Frédéric Valentin,et al.  Consistent Local Projection Stabilized Finite Element Methods , 2010, SIAM J. Numer. Anal..

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

[25]  Frédéric Valentin,et al.  Stabilized Finite Element Methods Based on Multiscale Enrichment for the Stokes Problem , 2006, SIAM J. Numer. Anal..

[26]  Frédéric Valentin,et al.  Towards multiscale functions : enriching finite element spaces with local but not bubble-like functions , 2005 .

[27]  Arif Masud,et al.  A multiscale finite element method for the incompressible Navier-Stokes equations , 2006 .

[28]  Alessandro Russo,et al.  Approximation of the Stokes problem by residual-free macro bubbles , 1996 .