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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 $$\\frac{p}{p+1}$$pp+1 Padé approximant of the exponential function. We show that for a uniform mesh of N elements there exist $$(p+1)N$$(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 $$2p+1$$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.

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