Stabilized formulation for the advection–diffusion–absorption equation using finite calculus and linear finite elements

A stabilized finite element method (FEM) for the 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. The basis of the method is detailed for the 1D problem. It is shown that the stabilization terms act as a non-linear additional diffusion governed by a single stabilization parameter. A critical constant value of this parameter ensuring a stabilized solution using two node linear elements is given. More accurate numerical results can be obtained by using a simple expression of the non-linear stabilization parameter depending on the signs of the numerical solution and of its derivatives. A straightforward two steps algorithm yielding a stable and accurate solution for all the range of physical parameters and boundary conditions is described. The extension to the multi-dimensional case is briefly described. Numerical results for 1D and 2D problems are presented showing the efficiency and accuracy of the new stabilized formulation.