{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,20]],"date-time":"2026-09-20T04:41:44Z","timestamp":1789879304233,"version":"4.0.1"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.1137\/110828010","type":"journal-article","created":{"date-parts":[[2012,6,28]],"date-time":"2012-06-28T14:35:51Z","timestamp":1340894151000},"page":"1801-1826","source":"Crossref","is-referenced-by-count":25,"title":["Analysis of an $hp$-Nonconforming Discontinuous Galerkin Spectral Element Method for Wave Propagation"],"prefix":"10.1137","volume":"50","author":[{"given":"Tan","family":"Bui-Thanh","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Omar","family":"Ghattas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,6,28]]},"reference":[{"key":"atypb1","volume-title":"Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables","author":"Abramowitz M.","year":"1965"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/1987210201991"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/0724049"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/1036141"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(98)00359-4"},{"key":"atypb6","first-page":"13","volume-title":"Nonlinear Partial Differential Equations and Their Applications","author":"Bernardi C.","year":"1994"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-30726-6"},{"key":"atypb9","first-page":"545","volume":"54","author":"Cockburn B.","year":"1990","journal-title":"Math. 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