{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:42:18Z","timestamp":1776724938020,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"229","license":[{"start":{"date-parts":[[2000,8,18]],"date-time":"2000-08-18T00:00:00Z","timestamp":966556800000},"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>\n                    In this paper we shall derive\n                    <italic>a posteriori<\/italic>\n                    error estimates in the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions. This result is mainly based on some fundamental a priori error estimates published in a recent paper by C. Chainais-Hillairet. The theoretical results are confirmed by numerical experiments.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01158-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"25-39","source":"Crossref","is-referenced-by-count":77,"title":["A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions"],"prefix":"10.1090","volume":"69","author":[{"given":"Dietmar","family":"Kr\u00f6ner","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mario","family":"Ohlberger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,8,18]]},"reference":[{"key":"1","unstructured":"C. Chainais-Hillairet: Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimates. Preprint http:\/\/umpa.ens-lyon.fr\/UMPA\/Prepublications\/0205\/, ENS Lyon (1997)."},{"key":"2","unstructured":"Cockburn,B. Gremaud, P.A.: An error estimate for finite element methods for scalar conservation laws. IMA Preprint Series No.1144 (1993)."},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"522","DOI":"10.1137\/0733028","article-title":"Error estimates for finite element methods for scalar conservation laws","volume":"33","author":"Cockburn, Bernardo","year":"1996","journal-title":"SIAM J. Numer. 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Oxford University Computing Laboratory, Numerical Analysis Group, Report Nr. 95\/24."},{"issue":"3","key":"10","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1002\/cpa.3160480302","article-title":"Adaptive finite element methods for conservation laws based on a posteriori error estimates","volume":"48","author":"Johnson, Claes","year":"1995","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1002\/(sici)1098-2426(199701)13:1<57::aid-num5>3.0.co;2-l","article-title":"The study on the nonlinear computations of the DQ and DC methods","volume":"13","author":"Chen, Wen","year":"1997","journal-title":"Numer. 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Oxford University Computing Laboratory, Numerical Analysis Group, Report Nr. 95\/01."},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1007\/BF02431999","article-title":"Convergence of higher order finite volume schemes on irregular grids","volume":"3","author":"Noelle, Sebastian","year":"1995","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"16","doi-asserted-by":"crossref","unstructured":"S\u00fcli, E., Houston, P.: Finite element methods for hyperbolic problems: aposteriori analysis and adaptivity. In: Duff, I. S. (ed.) et al., The state of the art in numerical analysis, Clarendon Press, Oxford (1997), 441-471.","DOI":"10.1093\/oso\/9780198500148.003.0017"},{"issue":"4","key":"17","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1137\/0728048","article-title":"Local error estimates for discontinuous solutions of nonlinear hyperbolic equations","volume":"28","author":"Tadmor, Eitan","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"18","doi-asserted-by":"publisher","first-page":"136","DOI":"10.1137\/S0036141094263755","article-title":"On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term","volume":"28","author":"Tveito, Aslak","year":"1997","journal-title":"SIAM J. Math. 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