Stabilized solution of the multidimensional advection–diffusion–absorption equation using linear finite elements

A stabilized finite element method (FEM) for the multidimensional steady state advection–diffusion–absorption equation is presented. The stabilized formulation is based on the modified governing differential equations derived via the finite calculus (FIC) method. For 1D problems the stabilization terms act as a nonlinear additional diffusion governed by a single stabilization parameter. It is shown that for multidimensional problems an orthotropic stabilizing diffusion must be added along the principal directions of curvature of the solution. A simple iterative algorithm yielding a stable and accurate solution for all the range of physical parameters and boundary conditions is described. Numerical results for 1D and 2D problems with sharp gradients are presented showing the effectiveness and accuracy of the new stabilized formulation.