On the Stability of Continuous-Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems

TL;DR

This paper introduces a coupled finite element method combining continuous and discontinuous Galerkin techniques for advection-diffusion-reaction problems, demonstrating stability under certain flow conditions and supported by numerical experiments.

Contribution

It presents a novel coupled Galerkin method that ensures stability in the streamline diffusion norm for problems with internal and boundary layers.

Findings

The coupled method is stable when the convection flows from continuous to discontinuous regions.

Numerical experiments confirm the theoretical stability results.

The approach effectively handles internal and boundary layers in advection-diffusion-reaction problems.

Abstract

We consider a finite element method which couples the continuous Galerkin method away from internal and boundary layers with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. The stability properties of the coupled method are illustrated by numerical experiments.