Stabilized finite element methods and feedback control for Burgers´ equation

The viscous Burgers´ equation is an example of an equation that has unstable Galerkin finite element approximations for very small viscosity coefficients, κ. The closed loop Galerkin approximation of a controlled Burgers´ equation is more stable than the open loop, but still unstable for very small κ. A well-known fix for the open loop simulation is the Galerkin-least-squares (GLS) method which adds stabilizing terms to the Galerkin method. The GLS method is stable for open loop simulations and closed loop simulations implemented with the linear feedback control law designed for the non-stabilized problem