{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:18:57Z","timestamp":1776795537001,"version":"3.51.2"},"reference-count":16,"publisher":"American Mathematical Society (AMS)","issue":"281","license":[{"start":{"date-parts":[[2013,6,5]],"date-time":"2013-06-05T00:00:00Z","timestamp":1370390400000},"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                    In this paper, we obtain the first local a posteriori error estimate for time-dependent Hamilton-Jacobi equations. Given an arbitrary domain\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Omega\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u03a9\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Omega<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and a time\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T\">\n                        <mml:semantics>\n                          <mml:mi>T<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">T<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the estimate gives an upper bound for the\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\">\n                        <mml:semantics>\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:annotation encoding=\"application\/x-tex\">L^\\infty<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -norm in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Omega\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u03a9\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Omega<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    at time\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T\">\n                        <mml:semantics>\n                          <mml:mi>T<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">T<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the difference between the viscosity solution\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u\">\n                        <mml:semantics>\n                          <mml:mi>u<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">u<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and any continuous function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"v\">\n                        <mml:semantics>\n                          <mml:mi>v<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">v<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in terms of the initial error in the domain of dependence and in terms of the (shifted) residual of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"v\">\n                        <mml:semantics>\n                          <mml:mi>v<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">v<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the union of all the cones of dependence with vertices in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Omega\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u03a9\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Omega<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The estimate holds for general Hamiltonians and any space dimension. It is thus an ideal tool for devising adaptive algorithms with rigorous error control for time-dependent Hamilton-Jacobi equations. This result is an extension to the global a posteriori error estimate obtained by S. Albert, B. Cockburn, D. French, and T. Peterson in\n                    <italic>A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part II: The time-dependent case<\/italic>\n                    , Finite Volumes for Complex Applications, vol. III, June 2002, pp. 17\u201324. Numerical experiments investigating the sharpness of the a posteriori error estimates are given.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2012-02610-x","type":"journal-article","created":{"date-parts":[[2012,6,5]],"date-time":"2012-06-05T14:27:03Z","timestamp":1338906423000},"page":"187-212","source":"Crossref","is-referenced-by-count":2,"title":["Local a posteriori error estimates for time-dependent Hamilton-Jacobi equations"],"prefix":"10.1090","volume":"82","author":[{"given":"Bernardo","family":"Cockburn","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ivan","family":"Merev","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianliang","family":"Qian","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2012,6,5]]},"reference":[{"issue":"12","key":"1","doi-asserted-by":"publisher","first-page":"1339","DOI":"10.1002\/(SICI)1097-0312(199612)49:12<1339::AID-CPA5>3.0.CO;2-B","article-title":"Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes","volume":"49","author":"Abgrall, R.","year":"1996","journal-title":"Comm. 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