{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,26]],"date-time":"2026-08-26T01:17:10Z","timestamp":1787707030381,"version":"build-2784847793"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>An anisotropic a posteriori error estimate is derived for a finite element discretization of the wave equation in two space dimensions. Only the error due to space discretization is considered, and the error estimates are derived in the nonnatural $L^2(0,T;H^1(\\Omega))$ norm using elliptic reconstruction. A numerical study of the effectivity index on unstructured, nonadapted, anisotropic meshes confirms the sharpness of the error estimator, provided the error due to time discretization is negligible compared to the finite element error. An anisotropic, adaptive finite element algorithm is then presented to control the finite element error in the $L^2(0,T;H^1(\\Omega))$ norm. Numerical results on adapted meshes indicate that the error estimator slightly underestimates the true error. We conjecture that the missing information corresponds to the interpolation error between successive meshes. It is observed that the error estimator becomes sharp again when considering the damped wave equation \\[\\varepsilon\\frac{\\partial^{2}u}{\\partial t^{2}}+\\frac{\\partial u}{\\partial t}-\\Delta u=f,\\] with small values of $\\varepsilon$, that is, when the parabolic character of the PDE becomes predominant.<\/jats:p>","DOI":"10.1137\/090778249","type":"journal-article","created":{"date-parts":[[2010,7,29]],"date-time":"2010-07-29T19:05:29Z","timestamp":1280430329000},"page":"2213-2234","source":"Crossref","is-referenced-by-count":13,"title":["Numerical Study of an Anisotropic Error Estimator in the $L^2(H^1)$ Norm for the Finite Element Discretization of the Wave Equation"],"prefix":"10.1137","volume":"32","author":[{"given":"Marco","family":"Picasso","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,7,29]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(02)00400-0"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2005.08.003"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620280912"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/090756995"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-05-01800-4"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2006.08.012"},{"key":"R7","unstructured":"F. Alauzet and M. Mehrenberger,\n                      P\n                      1\n                      -conservative Solution Interpolation on Unstructured Triangular Meshes\n                      , Rapport de Recherche 8604, Institut National de Recherche en Informatique et Automatique (INRIA), Rocquencourt, France, 2009."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.2478\/cmam-2010-0001"},{"key":"R9","first-page":"263","volume":"7","author":"Bangerth W.","year":"1999","journal-title":"East-West J. Numer. Math."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1142\/S0218396X01000668"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.06.022"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202505000339"},{"key":"R13","unstructured":"H. Borouchaki and P. Laug,\n                      The BL\n                      2\n                      D Mesh Generator: Beginner's Guide, User's and Programmer's Manual\n                      , Technical report RT-0194, Institut National de Recherche en Informatique et Automatique (INRIA), Rocquencourt, France, 1996."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.4171\/IFB\/74"},{"key":"R15","first-page":"77","volume":"9","author":"Cl\u00e9ment P.","year":"1975","journal-title":"RAIRO Anal. Num\u00e9r.","ISSN":"https:\/\/id.crossref.org\/issn\/0399-0516","issn-type":"print"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/0732033"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100273"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-002-0415-z"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/BF01401041"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/s002110000170"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1007\/PL00005407"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"O. Lakkis and C. Makridakis,\n                      Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems\n                      , Math. Comp., 75 (2006), pp. 1627\u20131658.","DOI":"10.1090\/S0025-5718-06-01858-8"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(97)00207-7"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"J.L. Lions and E. Magenes,\n                      Non-homogeneous Boundary Value Problems and Applications\n                      , Vol. I, Springer-Verlag, New York, 1972 (in French).","DOI":"10.1007\/978-3-642-65217-2"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/080715135"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902406314"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902403759"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827501398578"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1002\/cnm.546"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2005.11.018"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2009.09.004"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1002\/num.1690100509"},{"key":"R33","doi-asserted-by":"crossref","unstructured":"J. Stoer and R. Bulirsch,\n                      Introduction to Numerical Analysis\n                      , 3rd ed., Texts Appl. Math. 42, Springer-Verlag, New York, 2002 (in German).","DOI":"10.1007\/978-0-387-21738-3"},{"key":"R34","unstructured":"W. L. Wood,\n                      Practical Time-Stepping Schemes\n                      . Oxford Appl. Math. Comput. Sci. Ser. Clarendon Press, Oxford University Press, New York, 1990."},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620240206"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620330702"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090778249","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:49:30Z","timestamp":1787330970000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090778249"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":36,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090778249"],"URL":"https:\/\/doi.org\/10.1137\/090778249","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}