An analysis of classical techniques for consistent stabilisation of the advection-diffusion-reaction equation finite element solution

We analyse and study instability problems related to the solution of the advection-diffusion-reaction equation (ADR) using a standard finite element scheme. With this aim, this work has been carried out in the following way: first, three weak formulations are obtained from the general problem. In specific, we study the existence and uniqueness of the solution for each of the aforementioned formulations. Second, we analyse the general theory of consistent stabilisation techniques for the ADR equation, which includes: streamline upwind/Petrov-Galerkin (SUPG), and Galerkin/least-squares (GLS). Third, we study and develop, for linear triangular elements, two of the most important subgrid-scale techniques, i.e. algebraic subgrid scale (ASGS), and orthogonal subgrid scale (OSS). This includes the study of an expression for a stabilisation parameter based on an ADR equation's Fourier analysis. Finally, as conclusion, all these stabilisation techniques are put in context with the SUPG technique for a better comparison as well as understanding of their underlying features for linear triangular elements.