Subgrid Stabilized Defect Correction Methods for the Navier-Stokes Equations

We consider the synthesis of a recent subgrid stabilization method with defect correction methods. The combination is particularly efficient and combines the best algorithmic features of each. We prove convergence of the method for a fixed number of corrections as the mesh size goes to zero and derive parameter scalings from the analysis. We also present some numerical tests which both verify the theoretical predictions and illustrate the method’s promise.

[1]  P. Hemker,et al.  Mixed defect correction iteration for the accurate solution of the convection diffusion equation , 1982 .

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

[3]  Béatrice Rivière,et al.  A Discontinuous Subgrid Eddy Viscosity Method for the Time-Dependent Navier-Stokes Equations , 2005, SIAM J. Numer. Anal..

[4]  Mihai Anitescu,et al.  IMPLICIT FOR LOCAL EFFECTS AND EXPLICIT FOR NONLOCAL EFFECTS IS UNCONDITIONALLY STABLE , 2004 .

[5]  Barry Koren,et al.  Defect correction and nonlinear multigrid for steady Euler equations , 1988 .

[6]  Hans-Görg Roos,et al.  The e-uniform convergence of a defect correction method on a Shishkin mesh , 2001 .

[7]  Hyesuk Lee.  Analysis of a defect correction method for viscoelastic fluid flow , 2004 .

[8]  E. Tadmor,et al.  Analysis of the spectral vanishing viscosity method for periodic conservation laws , 1989 .

[9]  Torsten Linß,et al.  Defect correction on Shishkin-type meshes , 2001, Numerical Algorithms.

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

[11]  J. Guermond.  Stabilization of Galerkin approximations of transport equations by subgrid modelling , 1999 .

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

[13]  G. Juncu.  A numerical study of steady viscous flow past a fluid sphere , 1999 .

[14]  Barry Koren,et al.  Multigrid, defect correction and upwind schemes for the steady Navier-Stokes equations , 1988 .

[15]  J. Schwartz.  Nonlinear Functional Analysis , 1969 .

[16]  H. Stetter.  The defect correction principle and discretization methods , 1978 .

[17]  Owe Axelsson,et al.  Adaptive refinement for convection-diffusion problems based on a defect-correction technique and finite difference method , 1997, Computing.

[18]  W. Layton,et al.  A two-level variational multiscale method for convection-dominated convection-diffusion equations , 2006 .

[19]  P. Hemker.  An accurate method without directional bias for the numerical solution of a 2-D elliptic singular perturbation problem , 1982 .

[20]  R.M.M. Mattheij,et al.  Local defect correction with different grid types , 2002 .

[21]  An assessment of the finite termination property of the defect correction method , 1993 .

[22]  R.M.M. Mattheij,et al.  Local defect correction with slanting grids , 2002 .

[23]  Jean-Antoine Désidéri,et al.  Convergence Analysis of the Defect-Correction Iteration for Hyperbolic Problems , 1995, SIAM J. Sci. Comput..

[24]  O. Axelsson,et al.  Defect correction methods for convection dominated convection-diffusion problems , 1990 .

[25]  Grigorii I. Shishkin,et al.  The Use of Defect Correction for the Solution of Parabolic Singular Perturbation Problems , 1995 .

[26]  William Layton,et al.  Adaptive defect correction methods for convection dominated, convection diffusion problems , 2000 .

[27]  Irfan Altas,et al.  A high accuracy defect-correction multigrid method for the steady incompressible Navier-Stokes equations , 1994 .

[28]  William J. Layton,et al.  Adaptive Defect-Correction Methods for Viscous Incompressible Flow Problems , 2000, SIAM J. Numer. Anal..

[29]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[30]  Vivette Girault,et al.  Finite element approximation of the Navier-Stockes equations , 1979 .

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

[32]  K. Nechvíle.  The High Resolution , 2005 .

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

[34]  Wilhelm Heinrichs,et al.  Defect Correction for Convection-Dominated Flow , 1996, SIAM J. Sci. Comput..