Discontinuous Galerkin methods for fractional elliptic problems
Provides rigorous theoretical analysis for DG methods in fractional elliptic problems, which is incremental for the numerical analysis community.
The paper develops a mathematical framework for discontinuous Galerkin methods applied to 2D fractional elliptic problems, proving boundedness, stability, and optimal error estimates under energy and L2 norms, with numerical examples confirming optimal convergence.
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.