A Bubble Element for the Compressible Euler Equations

In this paper, we present a new Galerkin finite element method with bubble function for the compressible Euler equations. This method is derived from the scaled bubble element for the advection-diffusion problems developed by Simo and his colleagues, which is based on the equivalence between the Galerkin method employing piecewise linear interpolation with bubble functions and the Streamline-Upwind/Petrov Galerkin (SUPG) finite element method using P₁ approximation in the steady advection-diffusion problem. Simo and this author have applied this approach to transient advection-diffusion problems by using a special scaled bubble function called P-scaled bubble, which is designed to work in the transient advection-diffusion problems for any Peclet number from 0 to ∞. The method presented in this paper is an application of this p-scaled bubble element to a pure hyperbolic system.