The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

TL;DR

This paper develops and analyzes discontinuous Galerkin methods for fractional degenerate convection-diffusion equations, providing stability, convergence proofs, and numerical illustrations of solution behavior.

Contribution

It introduces a novel DG method tailored for fractional degenerate convection-diffusion equations and establishes its stability and convergence properties.

Findings

Proved stability estimates for the proposed methods.

Established convergence to entropy solutions.

Numerical experiments illustrate solution behaviors.

Abstract

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'evy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments.