{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,20]],"date-time":"2026-07-20T13:55:03Z","timestamp":1784555703257,"version":"3.55.0"},"reference-count":47,"publisher":"American Mathematical Society (AMS)","issue":"302","license":[{"start":{"date-parts":[[2017,2,15]],"date-time":"2017-02-15T00:00:00Z","timestamp":1487116800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this article we propose a unified analysis for conforming and non-conforming finite element methods that provides a partial answer to the problem of preserving discrete divergence constraints when computing numerical solutions to the time-dependent Maxwell system. In particular, we formulate a compatibility condition relative to the preservation of genuinely oscillating modes that takes the form of a generalized commuting diagram, and we show that compatible schemes satisfy convergence estimates leading to long-time stability with respect to stationary solutions. These findings are applied by specifying compatible formulations for several classes of Galerkin methods, such as the usual curl-conforming finite elements and the centered discontinuous Galerkin (DG) scheme. We also propose a new conforming\/non-conforming Galerkin (Conga) method where fully discontinuous solutions are computed by embedding the general structure of curl-conforming finite elements into larger DG spaces. In addition to naturally preserving one of the Gauss laws in a strong sense, the Conga method is both spectrally correct and energy conserving, unlike existing DG discretizations where the introduction of a dissipative penalty term is needed to avoid the presence of spurious modes.<\/p>","DOI":"10.1090\/mcom\/3079","type":"journal-article","created":{"date-parts":[[2015,6,3]],"date-time":"2015-06-03T10:28:05Z","timestamp":1433327285000},"page":"2651-2685","source":"Crossref","is-referenced-by-count":22,"title":["Gauss-compatible Galerkin schemes for time-dependent Maxwell equations"],"prefix":"10.1090","volume":"85","author":[{"given":"Martin","family":"Campos Pinto","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eric","family":"Sonnendr\u00fccker","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2016,2,15]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1016\/0045-7825(80)90004-3","article-title":"Study of an implicit scheme for integrating Maxwell\u2019s equations","volume":"22","author":"Adam, J. 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