Discontinuous Galerkin methods for fractional elliptic problems

The aim of this paper is to 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 L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document} norm are obtained for DG methods of the different formulations. Finally, the performed numerical examples confirm the optimal convergence order of the different formulations.