Dispersion Analysis for Discontinuous Spectral Element Methods

A framework for studying the amplitude and phase errors for discontinuous spectral element methods applied to wave propagation problems is presented. In this framework, boundary conditions can be accounted for and the spatial distribution of the errors within individual elements can be obtained. This is of importance for spectral element discretizations, for which it might be convenient to have the element size larger than the wavelength. When applied to multiple element discretizations, this allows identification of criteria for reducing the errors. While these criteria depend in general on the particular application and the discretization itself, an attempt is made to obtain optimal methods for the case when the wave propagation takes place over a large number of elements. Unfortunately, such optimization leads in the most general case to full mass matrices, and hence is useful mainly for linear problems.