Date:

Monday, October 7, 2002

Speaker:

Pavel Bochev

Affiliation:

Sandia National Laboratories

e-mail:

pbboche@sandia.gov

Title:

Experiences with stabilized FEM for the Stokes Problem

Abstract:

Stable and accurate finite element solution of the Stokes problem requires pairs of velocity and pressure spaces that satisfy the inf-sup compatibility condition. % In this talk we will focus on regularization algorithms for the mixed variational problem that lead to the class of consistently stabilized mixed Galerkin methods. Stabilized methods do not require the inf-sup condition and their solution is stable for any pair of conforming pressure-velocity spaces. %

However, stabilization leads to a dichotomy in the type of the variational equations - they are either weakly coercive and possibly symmetric or strongly coercive and non-symmetric. In addition, some techniques lead to unconditional stabilization while others give methods whose stability depends on proper choices of regularizing parameters and/or implementation. As a result, finite element algebraic problems generated by stabilized methods range from symmetric indefinite systems that are unconditionally stable to non-symmetric, conditionally positive definite problems.

% We will first present a comparative study of stabilized methods for the Stokes equations. We begin with a general framework, based on a penalized Lagrangian formulation, which can be used to generate the methods and to reveal their taxonomy. Then we focus on the impact of regularization parameters upon convergence and stability of finite element approximations and compare performance of iterative solvers.

Then we will consider a new pressure-Poisson stabilized formulation that has some very attractive properties. Our starting point is a continuous form that uses negative Sobolev space norms in the stabilization term. We show that this idealized form is unconditionally stable. However, the negative norm is not computable and so we only use this formulation as a template to define an alternative, discrete stabilized form. We prove that this form retains the unconditional stability of the idealized one, remains consistent for any choice of conforming finite elements, and yields optimal error estimates. The talk will conclude with numerical experiments that illustrate the unconditional stability of the new method.

This is joint work with T. Barth, J. Shadid (Sandia) and M. Gunzburger (Florida State).