### 2003-43

#### Finite Element Error Analysis and Implementation of a Variational Multiscale Method for the Navier-Stokes Equations

Series: 2003-43, Preprints

MSC:
65M60 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

Abstract:
The paper presents a variational multiscale method (VMS) for the
incompressible Navier-Stokes equations which is defined by a large
scale space $L^H$ for the velocity deformation tensor and a turbulent
viscosity $\nu_T$. The connection of this method to the standard
formulation of a VMS is explained. A finite element error analysis for
the velocity is presented. It is shown that the constants in the error
estimate, in particular in the dominating exponential factor, depend
in general on a reduced Reynolds number. It is studied under which
conditions on $L^H$, the VMS can be implemented easily and efficiently
into an existing finite element code for solving the Navier-Stokes
equations. Numerical tests with the Smagorinsky LES model
for $\nu_T$ are presented which show that the VMS behaves as expected
if $L^H$ is varied.

Keywords:
Variational multiscale method, finite element method, error analysis, Navier-Stokes equations