Implementation of the Continuous-Discontinuous Galerkin Finite Element Method

TL;DR

This paper discusses implementing a hybrid finite element method that combines the stability of discontinuous Galerkin with the efficiency of continuous methods for solving stationary advection-diffusion problems, using the deal.ii library.

Contribution

It presents the implementation details of a hybrid continuous-discontinuous Galerkin method and demonstrates its effectiveness through numerical experiments.

Findings

The hybrid method achieves stability comparable to discontinuous Galerkin.

Implementation using deal.ii is feasible and effective.

Numerical results validate the method's stability and efficiency.

Abstract

For the stationary advection-diffusion problem the standard continuous Galerkin method is unstable without some additional control on the mesh or method. The interior penalty discontinuous Galerkin method is stable but at the expense of an increased number of degrees of freedom. The hybrid method proposed in [5] combines the computational complexity of the continuous method with the stability of the discontinuous method without a significant increase in degrees of freedom. We discuss the implementation of this method using the finite element library deal.ii and present some numerical experiments.