{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T13:28:56Z","timestamp":1752672536167},"reference-count":13,"publisher":"University of Zielona G\u00f3ra, Poland","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2007,10,1]]},"abstract":"<jats:title>Arbitrary High-Order Finite Element Schemes and High-Order Mass Lumping<\/jats:title><jats:p>Computers are becoming sufficiently powerful to permit to numerically solve problems such as the wave equation with high-order methods. In this article we will consider Lagrange finite elements of order<jats:italic>k<\/jats:italic>and show how it is possible to automatically generate the mass and stiffness matrices of any order with the help of symbolic computation software. We compare two high-order time discretizations: an explicit one using a Taylor expansion in time (a Cauchy-Kowalewski procedure) and an implicit Runge-Kutta scheme. We also construct in a systematic way a high-order quadrature which is optimal in terms of the number of points, which enables the use of mass lumping, up to<jats:italic>P5<\/jats:italic>elements. We compare computational time and effort for several codes which are of high order in time and space and study their respective properties.<\/jats:p>","DOI":"10.2478\/v10006-007-0031-2","type":"journal-article","created":{"date-parts":[[2007,10,12]],"date-time":"2007-10-12T14:22:52Z","timestamp":1192198972000},"page":"375-393","source":"Crossref","is-referenced-by-count":20,"title":["Arbitrary High-Order Finite Element Schemes and High-Order Mass Lumping"],"prefix":"10.61822","volume":"17","author":[{"given":"S\u00e9bastien","family":"Jund","sequence":"first","affiliation":[]},{"given":"St\u00e9phanie","family":"Salmon","sequence":"additional","affiliation":[]}],"member":"37438","reference":[{"key":"1","doi-asserted-by":"crossref","DOI":"10.1002\/0470868279","author":"J Butcher","year":"2003","journal-title":"Numerical Methods for Ordinary Differential Equations"},{"key":"2","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1023\/A:1004420829610","article-title":"Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation","volume":"4","author":"M. 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Cohen","year":"1994","journal-title":"Finite Elements in Analysis and Design"},{"key":"7","first-page":"152","article-title":"Construction and analysis of higher order finite elements with mass lumping for the wave equation","author":"G. Cohen","year":"1993"},{"key":"8","unstructured":"Dumbser M. (2005): <i>Arbitrary High Order Schemes for the Solution of Hyperbolic Conservation Laws in Complex Domains<\/i>. Ph.D. thesis, Stuttgart University."},{"key":"9","first-page":"525","author":"G Fix","year":"1972","journal-title":"Effect of quadrature errors in the finite element approximation of steady state, eigenvalue and parabolic problems"},{"key":"10","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1002\/cpa.3160130205","article-title":"Systems of conservation laws","volume":"13","author":"P. Lax","year":"1960","journal-title":"Communications on Pure Applied Mathematics"},{"key":"11","first-page":"344","article-title":"A comparison between higher-order finite elements and finite differences for solving the wave equation","author":"W Mulder","year":"1996"},{"key":"12","unstructured":"Tordjman N. (1995): <i>El\u00e9ments finis d'ordre \u00e9lev\u00e9s avec condensation de masse pour l'\u00e9quation des onde.<\/i> Ph.D. Thesis, Universit\u00e9 Paris IX Dauphine, Paris."},{"key":"13","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1023\/A:1015126814947","article-title":"ADER: Arbitrary high order Godunov approach","volume":"1-4","author":"V. Titarev","year":"2002","journal-title":"Journal of Scientific Computing"}],"container-title":["International Journal of Applied Mathematics and Computer Science"],"original-title":[],"link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/amcs\/17\/3\/article-p375.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/view\/j\/amcs.2007.17.issue-3\/v10006-007-0031-2\/v10006-007-0031-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T10:26:23Z","timestamp":1709202383000},"score":1,"resource":{"primary":{"URL":"https:\/\/content.sciendo.com\/doi\/10.2478\/v10006-007-0031-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,10,1]]},"references-count":13,"journal-issue":{"issue":"3"},"URL":"https:\/\/doi.org\/10.2478\/v10006-007-0031-2","relation":{},"ISSN":["1641-876X"],"issn-type":[{"value":"1641-876X","type":"print"}],"subject":[],"published":{"date-parts":[[2007,10,1]]}}}