A Non-Linear Stabilized Finite Element Method with Shock Capturing and Reduced Smearing for Advection Problems: the Consistent Shock Capturing with Reduced Smearing Method

Abstract

In previous works the GPR method (Galerkin Projected Residual Method) was introduced. The GPR formulation has been applied with success to Helmholtz problem and to diffusion-reaction singularly perturbed problem. Based on the initial ideas of the GPR method, we developed the Galerkin Symmetrical Projected Residual Method (GSPR) for convection dominated diffusionconvection problems. The GSPR method is a linear finite element method with stabilization properties similar to SUPG method. However, for practical problems this method is not sufficiently stable and accurate.
In this work, we developed the new non-linear stabilized finite element method, based on the ideas of the shock-capturing stabilization. The method introduces new ideas about the upwind function and the stabilizing parameter τ . We observed that the stabilizing parameter is dependent on the degree of the interpolation polynomial, on the geometry of the element, on the advective field β , on the boundary conditions prescribed for the problem on Γ- and on the value of meas (Γe∩(Γ − Γ- )). A variety of upwind functions can be chosen to improve the spurious oscillations. The strategy to choose the stabilizing parameter is based on numerical experiment and on the requirements that overshooting and undershooting localized in narrow regions along sharp layers are not observed without leading to excessive smearing of the layers (CSCRS - Consistent Shock Capturing with Reduced Smearing Method). Some numerical tests for 2D problems are presented.