Numerical solutions of hyperbolic conservation laws: incorporating multi-resolution viscosity methods into the finite element framework

It is well known that the classic Galerkin finite-element method is unstable when applied to hyperbolic conservation laws, such as the Euler equations for compressible flow. Adding a diffusion term to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use. An elegant solution devised by Eitan Tadmor for spectral methods is to add diffusion only to the high frequency modes of the solution, which stabilizes the method without the sacrifice of accuracy. We incorporate this idea into the finite-element framework by using hierarchical functions as a multi-frequency basis. The result is a new finite element method for solving hyperbolic conservation laws. For this method, we are able to prove convergence for a one-dimensional scalar conservation Law; Numerical results are presented for one- and two-dimensional hyperbolic conservation laws.