{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:42:15Z","timestamp":1776724935843,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"229","license":[{"start":{"date-parts":[[2000,8,24]],"date-time":"2000-08-24T00:00:00Z","timestamp":967075200000},"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                    Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient. They are applied to a practical space-time discretization consisting of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C Superscript 0\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">C^{0}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    piecewise linear finite elements over highly graded unstructured meshes, and backward finite differences with varying time-steps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement\/coarsening. A simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach, and hints at the potentials and reliability of the proposed method.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01097-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"1-24","source":"Crossref","is-referenced-by-count":66,"title":["A posteriori error estimation and adaptivity for degenerate parabolic problems"],"prefix":"10.1090","volume":"69","author":[{"given":"R.","family":"Nochetto","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A.","family":"Schmidt","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C.","family":"Verdi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,8,24]]},"reference":[{"issue":"4","key":"1","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":"3","key":"2","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1016\/0899-8248(91)90006-G","article-title":"Local mesh refinement in 2 and 3 dimensions","volume":"3","author":"B\u00e4nsch, Eberhard","year":"1991","journal-title":"Impact Comput. Sci. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0899-8248","issn-type":"print"},{"key":"3","series-title":"Studies in Mathematics and its Applications, Vol. 4","isbn-type":"print","volume-title":"The finite element method for elliptic problems","author":"Ciarlet, Philippe G.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0444850287"},{"key":"4","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"},{"issue":"1","key":"5","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":"3","key":"6","doi-asserted-by":"publisher","first-page":"706","DOI":"10.1137\/0732033","article-title":"Adaptive finite element methods for parabolic problems. II. Optimal error estimates in \ud835\udc3f_{\u221e}\ud835\udc3f\u2082 and \ud835\udc3f_{\u221e}\ud835\udc3f_{\u221e}","volume":"32","author":"Eriksson, Kenneth","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"7","doi-asserted-by":"publisher","first-page":"1750","DOI":"10.1137\/0732079","article-title":"Adaptive finite element methods for parabolic problems. V. Long-time integration","volume":"32","author":"Eriksson, Kenneth","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"8","series-title":"A Wiley-Interscience Publication","isbn-type":"print","volume-title":"Variational principles and free-boundary problems","author":"Friedman, Avner","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0471868493"},{"key":"9","doi-asserted-by":"publisher","first-page":"567","DOI":"10.2307\/1968941","article-title":"On an inequality of Marcel Riesz","volume":"40","author":"Young, L. C.","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1007\/BF03167758","article-title":"Error estimates for multidimensional singular parabolic problems","volume":"4","author":"Nochetto, Ricardo H.","year":"1987","journal-title":"Japan J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0910-2043","issn-type":"print"},{"issue":"195","key":"11","doi-asserted-by":"publisher","first-page":"73","DOI":"10.2307\/2938664","article-title":"An adaptive finite element method for two-phase Stefan problems in two space dimensions. I. Stability and error estimates","volume":"57","author":"Nochetto, R. H.","year":"1991","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"5","key":"12","doi-asserted-by":"publisher","first-page":"1207","DOI":"10.1137\/0912065","article-title":"An adaptive finite element method for two-phase Stefan problems in two space dimensions. II. Implementation and numerical experiments","volume":"12","author":"Nochetto, R. H.","year":"1991","journal-title":"SIAM J. Sci. Statist. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0196-5204","issn-type":"print"},{"key":"13","unstructured":"R.H. Nochetto, A. Schmidt, and C. Verdi, Adapting meshes and time-steps for phase change problems, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. (9) Mat. Appl. 8 (1997), 273\u2013292."},{"key":"14","unstructured":"\\bysame, Adaptive solution of phase change problems over unstructured tetrahedral meshes, Grid Generation and Adaptive Algorithms (M. Luskin et al. eds.), Springer Verlag, New York (to appear)."},{"key":"15","first-page":"233","article-title":"The use of numerical integration in finite element methods for solving parabolic equations","author":"Raviart, P.-A.","year":"1973"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2000-69-229\/S0025-5718-99-01097-2\/S0025-5718-99-01097-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-229\/S0025-5718-99-01097-2\/S0025-5718-99-01097-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:11:34Z","timestamp":1776723094000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-229\/S0025-5718-99-01097-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,8,24]]},"references-count":15,"journal-issue":{"issue":"229","published-print":{"date-parts":[[2000,1]]}},"alternative-id":["S0025-5718-99-01097-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-01097-2","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":[[1999,8,24]]}}}