{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:42:17Z","timestamp":1776782537643,"version":"3.51.2"},"reference-count":38,"publisher":"American Mathematical Society (AMS)","issue":"246","license":[{"start":{"date-parts":[[2004,7,28]],"date-time":"2004-07-28T00:00:00Z","timestamp":1090972800000},"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                    We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varepsilon<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h\">\n                        <mml:semantics>\n                          <mml:mi>h<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and the time step size\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In particular, it is shown that all error bounds depend on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartFraction 1 Over epsilon EndFraction\">\n                        <mml:semantics>\n                          <mml:mfrac>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mi>\n                              \u03b5\n                              \n                            <\/mml:mi>\n                          <\/mml:mfrac>\n                          <mml:annotation encoding=\"application\/x-tex\">\\frac {1}{\\varepsilon }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    only in some lower polynomial order for small\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varepsilon<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen, and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.\n                  <\/p>","DOI":"10.1090\/s0025-5718-03-01588-6","type":"journal-article","created":{"date-parts":[[2004,4,21]],"date-time":"2004-04-21T12:29:08Z","timestamp":1082550548000},"page":"541-567","source":"Crossref","is-referenced-by-count":59,"title":["Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits"],"prefix":"10.1090","volume":"73","author":[{"given":"Xiaobing","family":"Feng","sequence":"first","affiliation":[]},{"given":"Andreas","family":"Prohl","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2003,7,28]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"issue":"2","key":"2","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/s0294-1449(16)30349-3","article-title":"On the singular limit in a phase field model of phase transitions","volume":"5","author":"Alikakos, Nicholas D.","year":"1988","journal-title":"Ann. Inst. H. Poincar\\'{e} Anal. Non Lin\\'{e}aire","ISSN":"https:\/\/id.crossref.org\/issn\/0294-1449","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1007\/BF00375025","article-title":"Convergence of the Cahn-Hilliard equation to the Hele-Shaw model","volume":"128","author":"Alikakos, Nicholas D.","year":"1994","journal-title":"Arch. Rational Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"S. Allen and J. W. Cahn. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall., 27:1084\u20131095, 1979.","DOI":"10.1016\/0001-6160(79)90196-2"},{"issue":"1-2","key":"5","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/S0167-2789(99)00109-8","article-title":"A phase-field model of solidification with convection","volume":"135","author":"Anderson, D. M.","year":"2000","journal-title":"Phys. D","ISSN":"https:\/\/id.crossref.org\/issn\/0167-2789","issn-type":"print"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0167-2789(96)00207-2","article-title":"Phase field models for hypercooled solidification","volume":"104","author":"Bates, P. W.","year":"1997","journal-title":"Phys. D","ISSN":"https:\/\/id.crossref.org\/issn\/0167-2789","issn-type":"print"},{"key":"7","isbn-type":"print","first-page":"1","article-title":"A phase-field model with a double obstacle potential","author":"Blowey, J. F.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3110138816"},{"key":"8","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4338-8","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0387941932"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/BF00254827","article-title":"An analysis of a phase field model of a free boundary","volume":"92","author":"Caginalp, Gunduz","year":"1986","journal-title":"Arch. Rational Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"issue":"11","key":"10","doi-asserted-by":"publisher","first-page":"5887","DOI":"10.1103\/PhysRevA.39.5887","article-title":"Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations","volume":"39","author":"Caginalp, G.","year":"1989","journal-title":"Phys. Rev. A (3)","ISSN":"https:\/\/id.crossref.org\/issn\/1050-2947","issn-type":"print"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1017\/S0956792598003520","article-title":"Convergence of the phase field model to its sharp interface limits","volume":"9","author":"Caginalp, Gunduz","year":"1998","journal-title":"European J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0956-7925","issn-type":"print"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1093\/imamat\/39.1.51","article-title":"A numerical analysis of an anisotropic phase field model","volume":"39","author":"Caginalp, Gunduz","year":"1987","journal-title":"IMA J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4960","issn-type":"print"},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1137\/0915007","article-title":"Phase field computations of single-needle crystals, crystal growth, and motion by mean curvature","volume":"15","author":"Caginalp, G.","year":"1994","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"14","doi-asserted-by":"crossref","unstructured":"J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys., 28:258\u2013267, 1958.","DOI":"10.1063\/1.1744102"},{"issue":"7-8","key":"15","doi-asserted-by":"publisher","first-page":"1371","DOI":"10.1080\/03605309408821057","article-title":"Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces","volume":"19","author":"Chen, Xinfu","year":"1994","journal-title":"Comm. Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0360-5302","issn-type":"print"},{"issue":"11-12","key":"16","doi-asserted-by":"publisher","first-page":"1705","DOI":"10.1080\/03605309608821243","article-title":"Existence, uniqueness, and regularity of classical solutions of the Mullins-Sekerka problem","volume":"21","author":"Chen, Xinfu","year":"1996","journal-title":"Comm. Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0360-5302","issn-type":"print"},{"issue":"2","key":"17","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1093\/imanum\/14.2.243","article-title":"An error estimate for a finite-element scheme for a phase field model","volume":"14","author":"Chen, Zhi Ming","year":"1994","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"key":"18","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":"19","doi-asserted-by":"crossref","unstructured":"J. B. Collins and H. Levine. Diffuse interface model of diffusion-limited crystal growth. Phys. Rev. B, 31:6119\u20136122, 1985.","DOI":"10.1103\/PhysRevB.31.6119"},{"issue":"5","key":"20","doi-asserted-by":"publisher","first-page":"1533","DOI":"10.2307\/2154960","article-title":"Geometrical evolution of developed interfaces","volume":"347","author":"de Mottoni, Piero","year":"1995","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"21","isbn-type":"print","first-page":"46","article-title":"Global existence and stability of solutions to the phase field equations","author":"Elliott, C. M.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3764324740"},{"issue":"9","key":"22","doi-asserted-by":"publisher","first-page":"1097","DOI":"10.1002\/cpa.3160450903","article-title":"Phase transitions and generalized motion by mean curvature","volume":"45","author":"Evans, L. C.","year":"1992","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"key":"23","doi-asserted-by":"crossref","unstructured":"X. Feng and A. Prohl. Numerical analysis of the Allen-Cahn equation and approximation of the mean curvature flows. Numer. Math., 94(1):33\u201365, 2003.","DOI":"10.1007\/s00211-002-0413-1"},{"key":"24","unstructured":"X. Feng and A. Prohl. Error Analysis of a Mixed Finite Element Method for the Cahn-Hilliard Equation, Numer. Math. (submitted), IMA-Preprint #1798, 2001."},{"key":"25","unstructured":"X. Feng and A. Prohl. Numerical Analysis of the Cahn-Hilliard Equation and Approximation for the Hele-Shaw problem, Interfaces and Free Boundaries (submitted), IMA-Preprint #1799, 2001."},{"key":"26","unstructured":"X. Feng and A. Prohl. Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interphase limits. IMA-Preprint #1817, 2001."},{"key":"27","first-page":"No. 48, 26","article-title":"Models for phase separation and their mathematics","author":"Fife, Paul C.","year":"2000","journal-title":"Electron. J. Differential Equations"},{"key":"28","series-title":"Research Notes in Mathematics","isbn-type":"print","volume-title":"Free boundary problems: theory and applications. Vol. I, II","volume":"78","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0273085891"},{"issue":"8","key":"29","doi-asserted-by":"publisher","first-page":"811","DOI":"10.1016\/0362-546X(88)90040-5","article-title":"Numerical simulations of nonlinear phase transitions. I. The isotropic case","volume":"12","author":"Fix, G. J.","year":"1988","journal-title":"Nonlinear Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0362-546X","issn-type":"print"},{"key":"30","isbn-type":"print","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1142\/9789814415309_0005","article-title":"Models of pattern formation in first-order phase transitions","author":"Langer, J. S.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/9971978423"},{"issue":"5","key":"31","doi-asserted-by":"publisher","first-page":"1015","DOI":"10.1137\/0725058","article-title":"The numerical analysis of a phase field model in moving boundary problems","volume":"25","author":"Lin, J. T.","year":"1988","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"32","doi-asserted-by":"crossref","unstructured":"G. B. McFadden, A. A. Wheeler, R. J. Braun, S. R. Coriell, and R. F. Sekerka. Phase-field models for anisotropic interfaces. Phys. Rev. E (3), 48(3):2016\u20132024, 1993.","DOI":"10.1103\/PhysRevE.48.2016"},{"key":"33","doi-asserted-by":"crossref","unstructured":"W. W. Mullins and J. Sekerka. Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Math., 34:322\u2013329, 1963.","DOI":"10.1063\/1.1702607"},{"key":"34","doi-asserted-by":"crossref","unstructured":"O. Penrose and P. C. Fife. On the relation between the standard phase-field model and a \u201cthermodynamically consistent\u201d phase-field model. Phys. D, 69(1-2):107\u2013113, 1993.","DOI":"10.1016\/0167-2789(93)90183-2"},{"issue":"1","key":"35","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1006\/jcph.1998.6122","article-title":"Adaptive mesh refinement computation of solidification microstructures using dynamic data structures","volume":"148","author":"Provatas, Nikolas","year":"1999","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"2","key":"36","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/BF00386194","article-title":"Convergence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling","volume":"131","author":"Soner, H. Mete","year":"1995","journal-title":"Arch. Rational Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"issue":"6","key":"37","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1017\/S0956792500002606","article-title":"A sharp interface limit of the phase field equations: one-dimensional and axisymmetric","volume":"7","author":"Stoth, Barbara E. E.","year":"1996","journal-title":"European J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0956-7925","issn-type":"print"},{"issue":"1","key":"38","first-page":"15","article-title":"Finite element analysis of the phase field model with nonsmooth initial data","volume":"19","author":"Yue, Xing Ye","year":"1996","journal-title":"Acta Math. Appl. Sinica","ISSN":"https:\/\/id.crossref.org\/issn\/0254-3079","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2004-73-246\/S0025-5718-03-01588-6\/S0025-5718-03-01588-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2004-73-246\/S0025-5718-03-01588-6\/S0025-5718-03-01588-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T13:43:38Z","timestamp":1776779018000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2004-73-246\/S0025-5718-03-01588-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,7,28]]},"references-count":38,"journal-issue":{"issue":"246","published-print":{"date-parts":[[2004,4]]}},"alternative-id":["S0025-5718-03-01588-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-03-01588-6","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":[[2003,7,28]]}}}