Stabilization methods for transient transport problems
Event details
| Date | 18.11.2009 |
| Hour | 16:15 |
| Speaker | Prof. Erik Burman |
| Location |
MAA110
|
| Category | Conferences - Seminars |
Stabilized finite element methods is by now a standard tool for the computation of hyperbolic transport equations or convection dominated convection--diffusion equations. The stationary problem is well understood and
both global and local estimates have been proven for several stabilized methods such as the Streamline Upwind Petrov Galerkin method (SUPG), the discontinuous Galerkin method (DG) and the Continuous Interior Penalty method (CIP), showing that these methods share similar properties.
In the transient case the situation is completely different. The standard method of lines treatment, consisting of first discretizing in space with finite elements and then in time using some standard finite dfference scheme, has resisted analysis when applied to the SUPG scheme, and it has been questioned if this method is stable. The DG and the CIP method on the other hand both are members of a class of methods with symmetric stabilization for which a method of lines treatment is straightforward. Several questions, however have remained open in this case as well, such as the unified analysis for diffusion-dominated and convection dominated flow and the analysis of explicit time stepping schemes.
In this talk we will first consider the case of the SUPG method and give a new analysis showing that, in the case of hyperbolic transport equations, the SUPG-method combined with standard A-stable time discretizations is stable and optimally convergent. The so called "small time-step instability" enters the analysis as a special case when smoothness of the exact solution is insufficient.
Then we will discuss symmetric stabilization methods and give some recent results showing that these methods allows a seamless transition from convection dominated flow to diffusion dominated flow with (quasi) optimality in all regimes both with respect to space and time.
Finally we will consider the case of explicit Runge-Kutta methods and present a new analysis of the second (RK2) and the third order (RK3) Runge-Kutta methods for linear symmetric hyperbolic systems unifying the DG and the CIP method. The key ingredient
here is new energy estimates for RK2 and RK3. The stabilization operator plays a crucial role for both stability and continuity in the resulting stability and error estimates.
Links
Practical information
- General public
- Free
Contact
- Prof. Jacques Rappaz