An edge based stabilized finite element method for solving compressible flows: Formulation and parallel implementation

This paper presents a finite element formulation for solving multidimensional compressible flows. This method is inspired by our experience with the SUPG, finite volume (FV) and discontinuous-Galerkin (DG) methods. Our objective is to obtain a stable and accurate finite element formulation for multi-dimensional hyperbolic-parabolic problems with particular emphasis on compressible flows. In the proposed formulation, the upwinding effect is introduced by considering the flow characteristics along the normal vectors to the element interfaces. This method is applied for solving inviscid, laminar and turbulent flows. The one-equation turbulence closure model of Spalart-Allmaras (S-A) is used. Several numerical tests are carried out, and a selection of two- and three-dimensional experiments is presented. The results are encouraging, and it is expected that more numerical experiments and theoretical analysis will lead to a greater insight into this formulation. We also discuss algorithmic and parallel implementation issues.