Analysis of an hp-Nonconforming Discontinuous Galerkin Spectral Element Method for Wave Propagation

We analyze the consistency, stability, and convergence of an hp discontinuous Galerkin spectral element method of Kopriva [J. Comput. Phys., 128 (1996), pp. 475–488] and Kopriva, Woodruff, and Hussaini [Internat. J. Numer. Methods Engrg., 53 (2002), pp. 105–122]. The analysis is carried out simultaneously for acoustic, elastic, coupled elastic-acoustic, and electromagnetic wave propagation. Our analytical results are developed for both conforming and nonconforming approximations on hexahedral meshes using either exact integration with Legendre–Gauss quadrature or inexact integration with Legendre–Gauss–Lobatto quadrature. A mortar-based nonconforming approximation is developed to treat both hand p-nonconforming meshes simultaneously. The mortar approach is constructed in such a way that consistency, stability, and convergence analyses for nonconforming approximations follow the conforming counterparts with minimal modifications. In particular, it casts hp-nonconforming interfaces into equivalent conforming faces on mortars, for which the analyses are easily carried out using standard approaches. Sharp hp-convergence results are then proved for nonconforming approximations of time-dependent wave propagation problems using both exact and inexact quadratures.