Approximation of the incompressible Navier-Stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes

Abstract This paper describes a finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales and a pressure segregation. The former consists of adding a least-square form of the component orthogonal to the finite element space of the convective and pressure gradient terms; this allows to deal with convection-dominated flows and to use equal velocity–pressure interpolation. The pressure segregation is inspired in fractional step schemes, although the converged solution corresponds to that of a monolithic time integration. Likewise, we put special emphasis on the use of anisotropic grids. In particular, we describe some possible choices for the calculation of the element length that appears in the stabilization parameters and check their behavior in two classical numerical examples. The preconditioning strategy used to solve the resulting algebraic system for very anisotropic meshes is also briefly described.

[1]  A. Quarteroni,et al.  Factorization methods for the numerical approximation of Navier-Stokes equations , 2000 .

[2]  R. Codina.  A stabilized finite element method for generalized stationary incompressible flows , 2001 .

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

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

[5]  David J. Silvester,et al.  Stabilised bilinear—constant velocity—pressure finite elements for the conjugate gradient solution of the Stokes problem , 1990 .

[6]  S. Turek.  a Comparative Study of Time-Stepping Techniques for the Incompressible Navier-Stokes Equations: from Fully Implicit Non-Linear Schemes to Semi-Implicit Projection Methods , 1996 .

[7]  Jim Douglas,et al.  An absolutely stabilized finite element method for the stokes problem , 1989 .

[8]  J. B. Perot,et al.  An analysis of the fractional step method , 1993 .

[9]  R. Codina.  Pressure Stability in Fractional Step Finite Element Methods for Incompressible Flows , 2001 .

[10]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II) , 1969 .

[11]  T. Hughes.  Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[12]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

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

[14]  Jie Shen,et al.  Hopf bifurcation of the unsteady regularized driven cavity flow , 1991 .

[15]  Ramesh Natarajan,et al.  A numerical method for incompressible viscous flow simulation , 1992 .

[16]  T. Apel,et al.  Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem , 1998 .

[17]  Pavel B. Bochev,et al.  Analysis of Velocity-Flux First-Order System Least-Squares Principles for the Navier--Stokes Equations: Part I , 1998 .

[18]  R. Codina.  Stabilized finite element approximation of transient incompressible flows using orthogonal subscales , 2002 .

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

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

[21]  A. Chorin.  A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[22]  Miguel Cervera,et al.  The intrinsic time for the streamline upwind/Petrov-Galerkin formulation using quadratic elements , 1992 .

[23]  Simona Perotto,et al.  Anisotropic Mesh Adaption with Application to CFD Problems , 2002 .

[24]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I) , 1969 .

[25]  Rainald Löhner,et al.  A linelet preconditioner for incompressible flow solvers , 2003 .

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

[27]  R. Codina,et al.  A numerical model to track two‐fluid interfaces based on a stabilized finite element method and the level set technique , 2002 .

[28]  Ramon Codina,et al.  Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations , 2000, Numerische Mathematik.

[29]  M. Fortin,et al.  Iterative stabilization of the bilinear velocity-constant pressure element , 1990 .

[30]  T. Hughes,et al.  Convergence analyses of Galerkin least-squares methods for symmetric advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations , 1993 .

[31]  Franco Brezzi,et al.  $b=\int g$ , 1997 .

[32]  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 .

[33]  R. Codina,et al.  A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation , 1997 .