Murtazo Nazarov
Division of Scientific Computing
Department of Information Technology
Uppsala University
Uppsala
Galerkin finite element approximation is known to be unstable for numerical approximation of convection dominated problems, especially for equations that model fluid motion. The problem becomes even more difficult when the speed of fluid gets close or exceeds the speed of sound. One of the approaches to overcome this issue is adding some additional terms to the finite element approximation. The so-called linear stabilizations such as Galerkin-Least-Squares (GLS, SUPG), General Galerkin (G2), edge viscosity, sub-grid viscosity are developed over the last three decades. It turns out that some linear stabilization techniques have adverse effects on nonlinear conservation laws with non-convex fluxes, i.e. the numerical solution converges to a weak non-entropic solution.
In this talk I summarize our recent developments in the field, specifically on nonlinear stabilization techniques.