Title: A multi-scale viscosity finite element method for hyperbolic conservation laws


			       Max Gunzburger
          School of Computational Science and Information Technology
                                CIST DSL/400
                         Florida State University
                        Tallahassee FL 32306-4120
                       Email: gunzburg@csit.fsu.edu


Abstract: (This is joint work with Marcus Calhoun-Lopez) 
The classic Galerkin finite element method is unstable when applied to 
hyperbolic conservation laws. Also, naively adding artificial diffusion to 
the equations stabilizes the method but sacrifices too much accuracy. An 
elegant approach, referred to as spectral viscosity methods, has been 
developed for spectral methods in which one adds diffusion only to the 
high-frequency modes of the solution, the result being that stabilization is 
effected without sacrificing accuracy. We extend this idea into the finite 
element framework by using hierarchical finite element functions as 
a multi-frequency basis. The result is a new finite element method for 
solving hyperbolic conservation laws in which artificial diffusion can be 
applied selectively only to the high-frequency modes of the approximation. 
We provide theoretical and computational verifications of the stability and 
accuracy of the method.