Discontinuous Galerkin methods for fractional elliptic problems

TL;DR

This paper develops and analyzes discontinuous Galerkin methods for solving 2D Riemann-Liouville fractional elliptic problems, providing stability, error estimates, and confirming optimal convergence through numerical examples.

Contribution

It introduces a mathematical framework for DG methods applied to fractional elliptic problems, including stability analysis and optimal error estimates.

Findings

DG methods are stable and bounded for fractional elliptic problems.

Optimal error estimates are achieved under energy and L2 norms.

Numerical examples confirm the theoretical convergence rates.

Abstract

We provide a mathematical framework for studying different versions of discontinuous Galerkin (DG) approaches for solving 2D Riemann-Liouville fractional elliptic problems on a finite domain. The boundedness and stability analysis of the primal bilinear form are provided. A priori error estimate under energy norm and optimal error estimate under $L^{2}$ norm are obtained for DG methods of the different formulations. Finally, the performed numerical examples confirm the optimal convergence order of the different formulations.