A finite element, multiresolution viscosity method for hyperbolic conservation laws

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. It is also well known that naively adding artificial diffusion to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use. 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 multifrequency 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. As for spectral viscosity methods, this results in stability without compromising accuracy. In the context of a one-dimensional scalar hyperbolic conservation law, we prove the convergence of approximate solutions, obtained using the new method, to the entropy solution of the conservation law. To illustrate the method, the results of a computational experiment for a one-dimensional hyperbolic conservation law are provided.