Stabilized Finite Elements in Fluid Dynamics for Simulation and Optimization

Prof. Dr. Malte Braack

Abstract

This lecture addresses finite elements for partial differential equations of saddle point structure and singularly perturbed elliptic problems. A common strategy to treat such problems with finite elements is to augment the discrete variational formulation by additional stabilization terms. Such additional terms should enhance the stability of the problem without sacrifying accuracy. This is performed by either
(i) strongly consistent stabilization terms, i.e. these terms vanish for the strong solution of the underlying problem if enough regularity is available, or
(ii) weakly consistent terms, i.e. the stabilization terms vanish for mesh size h -> 0 fast enough.

We present several strategies of finite element stabilization for convection-diffusion problems and for different flow problems (Stokes, Darcy, Navier-Stokes). In particular, we compare the accuracy in terms of a priori estimates, algorithmical consequences, and consequences in the context of optimal control problem. The types of stabilization cover methods of streamline diffusion type, local projection methods, interior penalty methods and residual free bubbles. Last but not least subgrid scale methods for turbulent Navier-Stokes equations are presented.

Date and Place

• 26.06.2014: 09:00 - 12:00, TUM MI 03.06.011
• 27.06.2014: 14:00 - 17:00, TUM MI 02.04.011
• 01.07.2014: 14:00 - 17:00, LRZ H.E.009, Leibniz-Rechenzentrum, Boltzmannstraße 1, Garching

We acknowledge the support of Leibniz Supercomputing Center (LRZ)