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

Abstract

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 $h$- and $p$-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.