{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:27:34Z","timestamp":1776842854160,"version":"3.51.2"},"reference-count":32,"publisher":"American Mathematical Society (AMS)","issue":"253","license":[{"start":{"date-parts":[[2006,9,29]],"date-time":"2006-09-29T00:00:00Z","timestamp":1159488000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We derive sharp\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript normal infinity Baseline left-parenthesis upper L Superscript 1 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L^\\infty (L^1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    a posteriori error estimates for initial boundary value problems of nonlinear convection-diffusion equations of the form\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartLayout 1st Row  StartFraction partial-differential u Over partial-differential t EndFraction plus d i v f left-parenthesis u right-parenthesis minus normal upper Delta upper A left-parenthesis u right-parenthesis equals g EndLayout\">\n                        <mml:semantics>\n                          <mml:mtable columnalign=\"right center left\" rowspacing=\"3pt\" columnspacing=\"0 thickmathspace\" side=\"left\" displaystyle=\"true\">\n                            <mml:mtr>\n                              <mml:mtd>\n                                <mml:mfrac>\n                                  <mml:mrow>\n                                    <mml:mi mathvariant=\"normal\">\n                                      \u2202\n                                      \n                                    <\/mml:mi>\n                                    <mml:mi>u<\/mml:mi>\n                                  <\/mml:mrow>\n                                  <mml:mrow>\n                                    <mml:mi mathvariant=\"normal\">\n                                      \u2202\n                                      \n                                    <\/mml:mi>\n                                    <mml:mi>t<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:mfrac>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mi>div<\/mml:mi>\n                                <mml:mo>\n                                  \u2061\n                                  \n                                <\/mml:mo>\n                                <mml:mi>f<\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>u<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mi mathvariant=\"normal\">\n                                  \u0394\n                                  \n                                <\/mml:mi>\n                                <mml:mi>A<\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>u<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mi>g<\/mml:mi>\n                              <\/mml:mtd>\n                            <\/mml:mtr>\n                          <\/mml:mtable>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{eqnarray*} \\frac {\\partial u}{\\partial t}+\\operatorname {div}f(u)-\\Delta A(u)=g \\end{eqnarray*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    under the nondegeneracy assumption\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A prime left-parenthesis s right-parenthesis greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>A<\/mml:mi>\n                              <mml:mo>\u2032<\/mml:mo>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">A\u2019(s)&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for any\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s element-of double-struck upper R\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s\\in \\mathbb {R}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The problem displays both parabolic and hyperbolic behavior in a way that depends on the solution itself. It is discretized implicitly in time via the method of characteristic and in space via continuous piecewise linear finite elements. The analysis is based on the Kru\u017ekov \u201cdoubling of variables\u201d device and the recently introduced \u201cboundary layer sequence\u201d technique to derive the entropy error inequality on bounded domains. The derived a posteriori error estimators have the correct convergence order in the region where the solution is smooth and recover the standard a posteriori error estimators known for parabolic equations with strong diffusions.\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01778-3","type":"journal-article","created":{"date-parts":[[2005,11,16]],"date-time":"2005-11-16T10:22:35Z","timestamp":1132136555000},"page":"43-71","source":"Crossref","is-referenced-by-count":7,"title":["Sharp \ud835\udc3f\u00b9 a posteriori error analysis for nonlinear convection-diffusion problems"],"prefix":"10.1090","volume":"75","author":[{"given":"Zhiming","family":"Chen","sequence":"first","affiliation":[]},{"given":"Guanghua","family":"Ji","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,9,29]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"404","DOI":"10.1137\/0732017","article-title":"A characteristics-mixed finite element method for advection-dominated transport problems","volume":"32","author":"Arbogast, Todd","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1007\/BF01176474","article-title":"Quasilinear elliptic-parabolic differential equations","volume":"183","author":"Alt, Hans Wilhelm","year":"1983","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"736","DOI":"10.1137\/0715049","article-title":"Error estimates for adaptive finite element computations","volume":"15","author":"Babu\u0161ka, I.","year":"1978","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1007\/s002050050152","article-title":"Entropy solutions for nonlinear degenerate problems","volume":"147","author":"Carrillo, Jos\u00e9","year":"1999","journal-title":"Arch. Ration. Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"issue":"6","key":"5","doi-asserted-by":"publisher","first-page":"1961","DOI":"10.1137\/S0036142998349102","article-title":"Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity","volume":"38","author":"Chen, Zhiming","year":"2001","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1137\/S1064827501383713","article-title":"On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients","volume":"24","author":"Chen, Zhiming","year":"2002","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"247","key":"7","doi-asserted-by":"publisher","first-page":"1167","DOI":"10.1090\/S0025-5718-04-01634-5","article-title":"An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems","volume":"73","author":"Chen, Zhiming","year":"2004","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1016\/S0045-7825(99)00295-9","article-title":"A characteristic Galerkin method with adaptive error control for the continuous casting problem","volume":"189","author":"Chen, Zhiming","year":"2000","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"9","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1051\/m2an\/197509R200771","article-title":"Approximation by finite element functions using local regularization","volume":"9","author":"Cl\u00e9ment, Ph.","year":"1975","journal-title":"Rev. Fran\\c{c}aise Automat. Informat. Recherche Op\\'{e}rationnelle S\\'{e}r. Rouge Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0397-9342","issn-type":"print"},{"key":"10","unstructured":"B. Cockburn, A simple introduction to error estimation for nonlinear hyperbolic conservation laws, Department of Mathematics, University of Minnesota, 2003. http:\/\/www.math.umn.edu\/ cockburn\/LectureNotes.html"},{"issue":"207","key":"11","doi-asserted-by":"publisher","first-page":"77","DOI":"10.2307\/2153563","article-title":"An error estimate for finite volume methods for multidimensional conservation laws","volume":"63","author":"Cockburn, Bernardo","year":"1994","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"12","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. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"13","doi-asserted-by":"publisher","first-page":"1487","DOI":"10.1137\/0726087","article-title":"Some improved error estimates for the modified method of characteristics","volume":"26","author":"Dawson, C. N.","year":"1989","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"14","doi-asserted-by":"publisher","first-page":"1106","DOI":"10.1137\/0733054","article-title":"A convergent adaptive algorithm for Poisson\u2019s equation","volume":"33","author":"D\u00f6rfler, Willy","year":"1996","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"15","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1007\/s002110050199","article-title":"A time- and space-adaptive algorithm for the linear time-dependent Schr\u00f6dinger equation","volume":"73","author":"D\u00f6rfler, Willy","year":"1996","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"5","key":"16","doi-asserted-by":"publisher","first-page":"871","DOI":"10.1137\/0719063","article-title":"Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures","volume":"19","author":"Douglas, Jim, Jr.","year":"1982","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"17","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1137\/0728003","article-title":"Adaptive finite element methods for parabolic problems. I. A linear model problem","volume":"28","author":"Eriksson, Kenneth","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"18","first-page":"419","article-title":"Error estimate for approximate solutions of a nonlinear convection-diffusion problem","volume":"7","author":"Eymard, Robert","year":"2002","journal-title":"Adv. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/1079-9389","issn-type":"print"},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/s002110100342","article-title":"Convergence of a finite volume scheme for nonlinear degenerate parabolic equations","volume":"92","author":"Eymard, Robert","year":"2002","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"3","key":"20","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":"233","key":"21","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1090\/S0025-5718-00-01187-X","article-title":"Adaptive Lagrange-Galerkin methods for unsteady convection-diffusion problems","volume":"70","author":"Houston, Paul","year":"2001","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"22","doi-asserted-by":"crossref","unstructured":"K. Huang, R. Zhang and M.T. van Genuchten, An Eulerian-Lagrangian approach with an adaptively corrected method of characteristic to simulate variably saturated water flow, Water Resource Research 30 (1994), 499-507.","DOI":"10.1029\/93WR02881"},{"issue":"3","key":"23","doi-asserted-by":"publisher","first-page":"858","DOI":"10.1137\/S0036142998336643","article-title":"Solution of degenerate convection-diffusion problems by the method of characteristics","volume":"39","author":"Ka\u010dur, J.","year":"2001","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5","key":"24","doi-asserted-by":"publisher","first-page":"1081","DOI":"10.3934\/dcds.2003.9.1081","article-title":"On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients","volume":"9","author":"Karlsen, Kenneth Hvistendahl","year":"2003","journal-title":"Discrete Contin. Dyn. Syst.","ISSN":"https:\/\/id.crossref.org\/issn\/1078-0947","issn-type":"print"},{"issue":"229","key":"25","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1090\/S0025-5718-99-01158-8","article-title":"A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions","volume":"69","author":"Kr\u00f6ner, Dietmar","year":"2000","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"26","first-page":"228","article-title":"First order quasilinear equations with several independent variables","volume":"81(123)","author":"Kru\u017ekov, S. N.","year":"1970","journal-title":"Mat. Sb. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0368-8666","issn-type":"print"},{"issue":"2","key":"27","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/s002050200184","article-title":"Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations","volume":"163","author":"Mascia, Corrado","year":"2002","journal-title":"Arch. Ration. Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"issue":"2","key":"28","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1051\/m2an:2001119","article-title":"A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations","volume":"35","author":"Ohlberger, Mario","year":"2001","journal-title":"M2AN Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"1","key":"29","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1006\/jdeq.1996.0155","article-title":"\ud835\udc3f\u00b9-contraction and uniqueness for quasilinear elliptic-parabolic equations","volume":"131","author":"Otto, Felix","year":"1996","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"issue":"3-4","key":"30","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1016\/S0045-7825(98)00121-2","article-title":"Adaptive finite elements for a linear parabolic problem","volume":"167","author":"Picasso, Marco","year":"1998","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"3","key":"31","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/BF01396435","article-title":"On the transport-diffusion algorithm and its applications to the Navier-Stokes equations","volume":"38","author":"Pironneau, O.","year":"1981","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"32","unstructured":"A. Schmidt and K.G. Siebert, ALBERT: An adaptive hierarchical finite element toolbox, IAM, University of Freiburg, 2000. http:\/\/www.mathematik.uni-freiburg.de\/IAM\/Research\/projectsdz\/albert."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-253\/S0025-5718-05-01778-3\/S0025-5718-05-01778-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-253\/S0025-5718-05-01778-3\/S0025-5718-05-01778-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:32:46Z","timestamp":1776781966000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-253\/S0025-5718-05-01778-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,9,29]]},"references-count":32,"journal-issue":{"issue":"253","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["S0025-5718-05-01778-3"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-05-01778-3","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":[[2005,9,29]]}}}