UNIVERSITY OF COLORADO AT DENVER

PLACE: Mathematics Conference Room 626 UCD Building, 1250 14th St., Denver

TIME: noon (Refreshments served at 11:45 am)

DATE: March 1,1999

ANALYSIS OF A STABILIZED FINITE ELEMENT METHOD FOR 
GENERALIZED INCOMPRESSIBLE NAVIER-STOKES EQUATIONS 
 
Ramon Codina

In this talk we discuss a finite element approximation of the incompressible
Navier-Stokes equations including zero order terms for the velocity, which
may arise from the Coriolis forces or when the permeability of the medium
needs to be taken into account.
 
The standard Galerkin method applied to this problem fails when the viscosity
is small because of several reasons. Apart from the pressure instabilities 
arising when equal velocity-pressure interpolations are used and the instabilities
produced by a dominant convective term, there are also global instabilities due
to the Coriolis forces and localized boundary layer instabilities due to permeability
effects. To circumvent all of them, a stabilized finite element method is considered,
which consists of adding to the Galerkin terms the element by element L2
product of the residual of the differential equations by the adjoint of the linearized
Navier-Stokes equations applied to the test functions and multiplied by suitable
numerical parameters.
 
The numerical analysis presented for the linearized problem reveals the role played
by the numerical parameters of the formulation, which allow to obtain a stable and
optimally accurate numerical formulation for all the ranges of the physical 
coefficients of the equations.
 
The contents of the talk are:
 
1. Problem statement
2. Stabilized finite element formulation
3. Analysis of the linearized problem
4. Some numerical results
5. Conclusions