On the suboptimality of the p-version discontinuous Galerkin methods for first order hyperbolic problems

TL;DR

This paper investigates the convergence rates of p-version discontinuous Galerkin methods for first order hyperbolic problems, revealing potential analysis limitations and demonstrating near-optimal convergence for specific convection fields.

Contribution

The paper identifies suboptimal convergence in existing analyses and provides refined results showing near-optimal rates for certain classes of convection fields.

Findings

Standard analysis shows 3/2 order suboptimality for general convection fields.

Numerical results do not show suboptimality, indicating analysis limitations.

For special convection fields, the method achieves only 1/2 order suboptimality.

Abstract

We address the issue of the suboptimality in the p-version discontinuous Galerkin (dG) methods for first order hyperbolic problems. The convergence rate is derived for the upwind dG scheme on tensor product meshes in any dimension. The standard proof in seminal work [14] leads to suboptimal convergence in terms of the polynomial degree by 3/2 order for general convection fields, with the exception of piecewise multi-linear convection fields, which rather yield optimal convergence. Such suboptimality is not observed numerically. Thus, it might be caused by a limitation of the analysis, which we partially overcome: for a special class of convection fields, we shall show that the dG method has a p-convergence rate suboptimal by 1/2 order only.