The efficient implementation of a finite element, multi-resolution 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 flows. Adding a diffusion term to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use. An elegant solution developed in the context of spectral methods by Eitan Tadmor and coworkers is to add diffusion only to the high frequency modes of the solution and can lead to stabilization without sacrificing 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, convergence for a one-dimensional scalar conservation law has previously been proved. Here, the method is described in detail, several issues connected with its efficient implementation are considered, and numerical results for several examples involving one- and two-dimensional hyperbolic conservation laws are provided. Several advantageous features of the method are discussed, including the ease for which discontinuities can be detected and artificial diffusion can be applied anisotropically and locally in physical as well as frequency space.