Finite element formulation for convective–diffusive problems with sharp gradients using finite calculus

Abstract

A finite element method (FEM) for steady-state convective–diffusive problems presenting sharp gradients of the solution both in the interior of the domain and in boundary layers is presented. The necessary stabilization of the numerical solution is provided by the Finite Calculus (FIC) approach. The FIC method is based in the solution by the Galerkin FEM of a modified set of governing equations which include characteristic length parameters. It is shown that the FIC balance equation for the multidimensional convection–diffusion problem written in the principal curvature axes of the solution, introduces an orthotropic diffusion which stabilizes the numerical solution both in smooth regions as well in the vicinity of sharp gradients. The dependence of the stabilization terms with the principal curvature directions of the solution makes the method non-linear. Details of the iterative scheme to obtain stabilized results are presented together with examples of application which show the efficiency and accuracy of the approach.