An analysis of the spectrum of the discontinuous Galerkin method II: Nonuniform grids

We analyze the eigenvalues of the discontinuous Galerkin spatial operator for the one-dimensional linear advection equation on nonuniform grids. We show that when the difference in cell sizes is below an order-dependent critical number, the spectrum continuously changes with changing cell sizes. In a particularly simple case where the mesh contains cells of only two sizes, the spectrum grows linearly with the proportion of smaller cells in the mesh. When the cell size scale is larger than a critical value, the eigenvalues corresponding to smaller cells separate from the rest of the spectrum. We provide an easily computable estimate of the upper bound on the spectrum for both cases. This estimate can be used to compute a relaxed time step restriction if smaller cells are dispersed among larger cells. Numerical examples for one- and two-dimensional problems reveal that computational saving can be realized by use of a larger stable time step.