Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes

TL;DR

This paper develops high-order nodal discontinuous Galerkin methods for solving fractional diffusion equations on 2D domains with triangular meshes, providing stability analysis, error estimates, and confirming optimal convergence through numerical experiments.

Contribution

The paper introduces a novel application of nodal discontinuous Galerkin methods with high-order basis functions for fractional diffusion equations on 2D triangular meshes, including stability and error analysis.

Findings

Methods achieve (N+1)-th order accuracy for degree N polynomials.

Numerical experiments confirm the optimal convergence order.

Stability and error estimates are established for the proposed schemes.

Abstract

This paper, as the sequel to previous work, develops numerical schemes for fractional diffusion equations on a two-dimensional finite domain with triangular meshes. We adopt the nodal discontinuous Galerkin methods for the full spatial discretization by the use of high-order nodal basis, employing multivariate Lagrange polynomials defined on the triangles. Stability analysis and error estimates are provided, which shows that if polynomials of degree $N$ are used, the methods are (N+1)-th order accurate for general triangulations. Finally, the performed numerical experiments confirm the optimal order of convergence.