An analysis of the spectrum of the discontinuous Galerkin method

We derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal [p/p+1] Padé approximation of exp(−z) when pth degree basis functions are used. We derive an upper bound on the eigenvalue with the largest magnitude as (p+1)(p+2). We demonstrate that this bound is not tight and prove that the asymptotic growth rate of the spectral radius is slower than quadratic in p. We also analyze the behavior of the spectrum near the imaginary axis to demonstrate that the spectral curves approach the imaginary axis although there are no purely imaginary eigenvalues.