Local Projection Stabilization for the Oseen System on Anisotropic Cartesian Meshes

Classical residual-based stabilization techniques, as for instance streamline upwind Petrov-Galerkin, as well as the local projection method are optimal on isotropic meshes. Here we extend the local projection stabilization for the Navier-Stokes system to anisotropic Cartesian meshes. We describe the new method and give an a priori error estimate for the two-dimensional case. The method leads on anisotropic meshes to qualitatively better convergence behavior than other isotropic stabilization methods. The capability of the method is illustrated by means of a numerical test problem.

[1]  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 .

[2]  Roland Becker,et al.  Parameter identification for chemical models in combustion problems , 2005 .

[3]  Peter Hansbo,et al.  A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equation , 1990 .

[4]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[5]  Erik Burman,et al.  Local Projection Stabilization for the Oseen Problem and its Interpretation as a Variational Multiscale Method , 2006, SIAM J. Numer. Anal..

[6]  Ralf Gritzki,et al.  Stabilized FEM with Anisotropic Mesh Refinement for the Oseen Problem , 2006 .

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

[8]  L. Formaggia,et al.  Anisotropic mesh adaptation in computational fluid dynamics: application to the advection-diffusion-reaction and the Stokes problems , 2004 .

[9]  Simona Perotto,et al.  Anisotropic error estimates for elliptic problems , 2003, Numerische Mathematik.

[10]  Thomas Richter,et al.  Local Projection Stabilization for the Stokes System on Anisotropic Quadrilateral Meshes , 2006 .

[11]  T. Richter,et al.  SOLUTIONS OF 3D NAVIER-STOKES BENCHMARK PROBLEMS WITH ADAPTIVE FINITE ELEMENTS , 2006 .

[12]  L. Franca,et al.  Stabilized finite element methods. II: The incompressible Navier-Stokes equations , 1992 .

[13]  R. Codina.  Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods , 2000 .

[14]  Torsten Linß.  Anisotropic meshes and streamline-diffusion stabilization for convection-diffusion problems , 2005 .

[15]  Lutz Tobiska,et al.  A modified streamline diffusion method for solving the stationary Navier-Stokes equation , 1991 .

[16]  Malte Braack,et al.  Stabilized finite elements for 3D reactive flows , 2006 .

[17]  T. Apel,et al.  Anisotropic mesh refinement in stabilized Galerkin methods , 1996 .