A Parallel Subgrid Stabilized Finite Element Method Based on Two-Grid Discretization for Simulation of 2D/3D Steady Incompressible Flows

Based on domain decomposition and two-grid discretization, a parallel subgrid stabilized finite element method for simulation of 2D/3D steady convection dominated incompressible flows is proposed and analyzed. In this method, a subgrid stabilized nonlinear Navier–Stokes problem is first solved on a coarse grid where the stabilization term is based on an elliptic projection defined on the same coarse grid, and then corrections are calculated in overlapped fine grid subdomains by solving a linearized problem. By the technical tool of local a priori estimate for finite element solution, error bounds of the approximate solution are estimated. Algorithmic parameter scalings of the method are derived. Numerical results are also given to demonstrate the effectiveness of the method.

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

[2]  Yueqiang Shang,et al.  A parallel two-level linearization method for incompressible flow problems , 2011, Appl. Math. Lett..

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

[4]  Yinnian He,et al.  Assessment of subgrid-scale models for the incompressible Navier-Stokes equations , 2010, J. Comput. Appl. Math..

[5]  Yinnian He,et al.  Newton Iterative Parallel Finite Element Algorithm for the Steady Navier-Stokes Equations , 2010, J. Sci. Comput..

[6]  Douglas N. Arnold,et al.  Local error estimates for finite element discretization of the Stokes equations , 1995 .

[7]  P. Hood,et al.  A numerical solution of the Navier-Stokes equations using the finite element technique , 1973 .

[8]  Volker John,et al.  A Finite Element Variational Multiscale Method for the Navier-Stokes Equations , 2005, SIAM J. Sci. Comput..

[9]  Sanjay Mittal,et al.  Parallel finite element computation of incompressible flows , 2009, Parallel Comput..

[10]  Xiaojun Zhou,et al.  A new parallel finite element algorithm for the stationary Navier-Stokes equations , 2011 .

[11]  Yinnian He,et al.  Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations☆ , 2009 .

[12]  Jinchao Xu,et al.  Local and Parallel Finite Element Algorithms Based on Two-Grid Discretizations for Nonlinear Problems , 2001, Adv. Comput. Math..

[13]  Jinchao Xu,et al.  Local and parallel finite element algorithms based on two-grid discretizations , 2000, Math. Comput..

[14]  Yinnian He,et al.  Parallel iterative finite element algorithms based on full domain partition for the stationary Navier--Stokes equations , 2010 .

[15]  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 .

[16]  John N. Shadid,et al.  A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations , 2008, J. Comput. Phys..

[17]  Dominik Obrist,et al.  High-order accurate solution of the incompressible Navier-Stokes equations on massively parallel computers , 2010, Journal of Computational Physics.

[18]  Béatrice Rivière,et al.  Subgrid Stabilized Defect Correction Methods for the Navier-Stokes Equations , 2006, SIAM J. Numer. Anal..

[19]  Yinnian He,et al.  A parallel Oseen-linearized algorithm for the stationary Navier-Stokes equations , 2012 .

[20]  Yueqiang Shang.  A parallel subgrid stabilized finite element method based on fully overlapping domain decomposition for the Navier–Stokes equations ☆ , 2013 .

[21]  W. Layton,et al.  A connection between subgrid scale eddy viscosity and mixed methods , 2002, Appl. Math. Comput..

[22]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[23]  Jean-Luc Guermond,et al.  Subgrid stabilized projection method for 2D unsteady flows at high Reynolds numbers , 2006 .

[24]  G. Burton.  Sobolev Spaces , 2013 .

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

[26]  Volker John,et al.  Numerical Studies of Finite Element Variational Multiscale Methods for Turbulent Flow Simulations , 2010 .

[27]  J. Spencer,et al.  EVOLUTION OF THE n-CUBE , 1979 .

[28]  Yi-chen Ma,et al.  Local and parallel finite element algorithms based on two-grid discretization for the stream function form of Navier-Stokes equations , 2006, Appl. Math. Comput..

[29]  马飞遥,et al.  Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations , 2007 .

[30]  Lois Mansfield,et al.  Finite element subspaces with optimal rates of convergence for the stationary Stokes problem , 1982 .

[31]  Jinchao Xu,et al.  Local and parallel finite element algorithms for the stokes problem , 2008, Numerische Mathematik.

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

[33]  E. Erturk,et al.  Numerical solutions of 2‐D steady incompressible driven cavity flow at high Reynolds numbers , 2004, ArXiv.

[34]  Béatrice Rivière,et al.  A two‐grid stabilization method for solving the steady‐state Navier‐Stokes equations , 2006 .

[35]  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 .

[36]  Yinnian He,et al.  A comparison of three kinds of local and parallel finite element algorithms based on two-grid discretizations for the stationary Navier–Stokes equations , 2011 .

[37]  Yueqiang Shang,et al.  A two-level subgrid stabilized Oseen iterative method for the steady Navier-Stokes equations , 2013, J. Comput. Phys..

[38]  A. H. Schatz,et al.  Interior estimates for Ritz-Galerkin methods , 1974 .

[39]  T. Hughes,et al.  Large Eddy Simulation and the variational multiscale method , 2000 .

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

[41]  W. Layton,et al.  A defect-correction method for the incompressible Navier-Stokes equations , 2002, Appl. Math. Comput..

[42]  L. Wahlbin,et al.  Interior maximum-norm estimates for finite element methods, part II , 1995 .

[43]  Yueqiang Shang.  A parallel two-level finite element variational multiscale method for the Navier–Stokes equations ☆ , 2013 .