We consider a finite element method for the nonhomogeneous second-order wave equation, which is formulated in terms of continuous approximation functions in both space and time, thereby giving a unified treatment of the spatial and temporal discretizations. Our analysis uses primarily energy arguments, which are quite common for spatial discretizations but not for time.<\/p>\n
We present a priori nodal (in time) superconvergence error estimates without any special time step restrictions. Our method is based on tensor-product spaces for the full discretization.<\/p>","DOI":"10.1090\/s0025-5718-96-00685-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"491-506","source":"Crossref","is-referenced-by-count":104,"title":["A continuous space-time finite element method for the wave equation"],"prefix":"10.1090","volume":"65","author":[{"given":"Donald","family":"French","sequence":"first","affiliation":[]},{"given":"Todd","family":"Peterson","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"issue":"186","key":"1","doi-asserted-by":"publisher","first-page":"255","DOI":"10.2307\/2008467","article-title":"Continuous finite elements in space and time for the heat equation","volume":"52","author":"Aziz, A. K.","year":"1989","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"2","unstructured":"I. Babu\u0161ka and T. Janik, The h-p version of the finite element method, SIAM J. Numer. Anal. 5 (1990), 363\u2013399."},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1051\/m2an\/1979130200751","article-title":"Semidiscrete and single step fully discrete approximations for second order hyperbolic equations","volume":"13","author":"Baker, Garth A.","year":"1979","journal-title":"RAIRO Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0399-0516","issn-type":"print"},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1016\/0898-1221(94)90048-5","article-title":"Continuous finite elements in space and time for the nonhomogeneous wave equation","volume":"27","author":"Bales, L.","year":"1994","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"issue":"209","key":"5","doi-asserted-by":"publisher","first-page":"89","DOI":"10.2307\/2153324","article-title":"Negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous \ud835\udc3f\u2082 Dirichlet boundary data","volume":"64","author":"Bales, L.","year":"1995","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"7","key":"6","first-page":"859","article-title":"Fully discrete Galerkin methods for the Korteweg-de Vries equation","volume":"12","author":"Bona, Jerry L.","year":"1986","journal-title":"Comput. Math. Appl. Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0886-9553","issn-type":"print"},{"key":"7","unstructured":"J. DeFrutos and J. M. Sanz-Serna, Accuracy and conservation properties in numerical integration: the case of the Korteweg-deVries equation, preprint."},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1137\/0724021","article-title":"Analysis of a continuous finite element method for hyperbolic equations","volume":"24","author":"Falk, Richard S.","year":"1987","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1093\/imanum\/14.3.421","article-title":"Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems","volume":"14","author":"French, Donald A.","year":"1994","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"key":"10","unstructured":"D. A. French and T. E. Peterson, A space-time finite element method for the second order wave equation, Report CMA-MR13-91, Centre for Mathematics and its Applications, Canberra."},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1016\/S0096-3003(20)80006-X","article-title":"Continuous finite element methods which preserve energy properties for nonlinear problems","volume":"39","author":"French, Donald A.","year":"1990","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"issue":"197","key":"12","doi-asserted-by":"publisher","first-page":"83","DOI":"10.2307\/2153022","article-title":"Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension","volume":"58","author":"Glassey, R. T.","year":"1992","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1016\/0377-0427(91)90022-C","article-title":"On the characterization of finite differences \u201coptimal\u201d meshes","volume":"36","author":"de Oliveira, Paula","year":"1991","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"1-2","key":"14","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1016\/0045-7825(93)90170-3","article-title":"Discontinuous Galerkin finite element methods for second order hyperbolic problems","volume":"107","author":"Johnson, Claes","year":"1993","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1137\/0730018","article-title":"On optimal order error estimates for the nonlinear Schr\u00f6dinger equation","volume":"30","author":"Karakashian, Ohannes","year":"1993","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"16","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5561-1","volume-title":"Semigroups of linear operators and applications to partial differential equations","volume":"44","author":"Pazy, A.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0387908455"},{"key":"17","unstructured":"G. R. Richter, An explicit finite element method for the wave equation, preprint."},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1016\/0021-9991(78)90038-4","article-title":"Numerical solution of a nonlinear Klein-Gordon equation","volume":"28","author":"Strauss, Walter","year":"1978","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00685-0\/S0025-5718-96-00685-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00685-0\/S0025-5718-96-00685-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:06:43Z","timestamp":1776719203000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00685-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":18,"journal-issue":{"issue":"214","published-print":{"date-parts":[[1996,4]]}},"alternative-id":["S0025-5718-96-00685-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00685-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1996]]}}}