Analysis of an interface stabilised finite element method: The advection-diffusion-reaction equation

TL;DR

This paper analyzes an interface stabilized finite element method for the advection-diffusion-reaction equation, demonstrating stability, convergence, and good numerical properties through theoretical proofs and numerical experiments.

Contribution

It provides a rigorous stability analysis and convergence proofs for the interface stabilized finite element method applied to advection-diffusion-reaction equations, with supporting numerical validation.

Findings

Proved stability via inf-sup and coercivity conditions.

Established convergence rates of order k+1/2 and k+1.

Demonstrated good stability and minimal dissipation in simulations.

Abstract

Analysis of an interface stabilised finite element method for the scalar advection-diffusion-reaction equation is presented. The method inherits attractive properties of both continuous and discontinuous Galerkin methods, namely the same number of global degrees of freedom as a continuous Galerkin method on a given mesh and the stability properties of discontinuous Galerkin methods for advection dominated problems. Simulations using the approach in other works demonstrated good stability properties with minimal numerical dissipation, and standard convergence rates for the lowest order elements were observed. In this work, stability of the formulation, in the form of an inf-sup condition for the hyperbolic limit and coercivity for the elliptic case, is proved, as is order $k+1/2$ order convergence for the advection-dominated case and order $k+1$ convergence for the diffusive limit in the $L^{2}$ norm. The analysis results are supported by a number of numerical experiments.