Spatial and Modal Superconvergence of the Discontinuous Galerkin Method for Linear Equations

TL;DR

This paper analyzes the superconvergence properties of the discontinuous Galerkin method for linear equations, revealing polynomial solutions related to Padé approximants and demonstrating exponential convergence of superconvergent points.

Contribution

It provides a new PDE-based analysis of DG superconvergence, linking polynomial solutions to Padé approximants and showing exponential convergence of superconvergent points.

Findings

Physical mode error is of order 2p+1.

Non-physical modes decay exponentially.

Superconvergent points rapidly approach Radau points.

Abstract

We apply the discontinuous Galerkin finite element method with a degree $p$ polynomial basis to the linear advection equation and derive a PDE which the numerical solution solves exactly. We use a Fourier approach to derive polynomial solutions to this PDE and show that the polynomials are closely related to the $\frac{p}{p+1}$ Pad\'e approximant of the exponential function. We show that for a uniform mesh of $N$ elements there exist $(p+1)N$ independent polynomial solutions, $N$ of which can be viewed as physical and $pN$ as non-physical. We show that the accumulation error of the physical mode is of order $2p+1$. In contrast, the non-physical modes are damped out exponentially quickly. We use these results to present a simple proof of the superconvergence of the DG method on uniform grids as well as show a connection between spatial superconvergence and the superaccuracies in dissipation and dispersion errors of the scheme. Finally, we show that for a class of initial projections on a uniform mesh, the superconvergent points of the numerical error tend exponentially quickly towards the downwind based Radau points.