Analysis of an hp-nonconforming discontinuous galerkin spectral element method for wave propagation

摘要

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.