{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:52:54Z","timestamp":1776793974765,"version":"3.51.2"},"reference-count":28,"publisher":"American Mathematical Society (AMS)","issue":"276","license":[{"start":{"date-parts":[[2012,4,14]],"date-time":"2012-04-14T00:00:00Z","timestamp":1334361600000},"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>A new method is proposed to improve the numerical simulation of time dependent problems when the initial and boundary data are not compatible. Unlike earlier methods limited to space dimension one, this method can be used for any space dimension. When both methods are applicable (in space dimension one), the improvements in precision are comparable, but the method proposed here is not restricted by dimension.<\/p>","DOI":"10.1090\/s0025-5718-2011-02469-5","type":"journal-article","created":{"date-parts":[[2011,4,14]],"date-time":"2011-04-14T08:04:26Z","timestamp":1302768266000},"page":"2071-2096","source":"Crossref","is-referenced-by-count":16,"title":["Treatment of incompatible initial and boundary data for parabolic equations in higher dimension"],"prefix":"10.1090","volume":"80","author":[{"given":"Qingshan","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhen","family":"Qin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Roger","family":"Temam","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2011,4,14]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"L.K. Bieniasz, A singularity correction procedure for digital simulation of potential-step chronoamperometric transients in one\u2013dimensional homogeneous reaction-diffusion systems, Electrochimica Acta 50 (2005), 3253\u20133261.","DOI":"10.1016\/j.electacta.2004.11.054"},{"issue":"3-4","key":"2","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/S0045-7825(98)00358-2","article-title":"Compatibility conditions for time-dependent partial differential equations and the rate of convergence of Chebyshev and Fourier spectral methods","volume":"175","author":"Boyd, John P.","year":"1999","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"3-4","key":"3","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/S0045-7825(98)00358-2","article-title":"Compatibility conditions for time-dependent partial differential equations and the rate of convergence of Chebyshev and Fourier spectral methods","volume":"175","author":"Boyd, John P.","year":"1999","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"4","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139086967","volume-title":"The one-dimensional heat equation","volume":"23","author":"Cannon, John Rozier","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0201135221"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"568","DOI":"10.4208\/cicp.110909.160310s","article-title":"Numerical resolution near \ud835\udc61=0 of nonlinear evolution equations in the presence of corner singularities in space dimension 1","volume":"9","author":"Chen, Qingshan","year":"2011","journal-title":"Commun. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/1815-2406","issn-type":"print"},{"key":"6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0002-9904-1943-07818-4","article-title":"Variational methods for the solution of problems of equilibrium and vibrations","volume":"49","author":"Courant, R.","year":"1943","journal-title":"Bull. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9904","issn-type":"print"},{"issue":"3","key":"7","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1002\/fld.1650020302","article-title":"The implementation of normal and\/or tangential boundary conditions in finite element codes for incompressible fluid flow","volume":"2","author":"Engelman, M. S.","year":"1982","journal-title":"Internat. J. Numer. Methods Fluids","ISSN":"https:\/\/id.crossref.org\/issn\/0271-2091","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"526","DOI":"10.1016\/S0021-9991(02)00034-7","article-title":"Accurate numerical resolution of transients in initial-boundary value problems for the heat equation","volume":"184","author":"Flyer, Natasha","year":"2003","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"546","DOI":"10.1137\/S0036139902415853","article-title":"On the nature of initial-boundary value solutions for dispersive equations","volume":"64","author":"Flyer, Natasha","year":"2003","journal-title":"SIAM J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1399","issn-type":"print"},{"issue":"5","key":"10","doi-asserted-by":"publisher","first-page":"1731","DOI":"10.1137\/S1064827500374169","article-title":"The convergence of spectral and finite difference methods for initial-boundary value problems","volume":"23","author":"Flyer, Natasha","year":"2002","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"3","key":"11","first-page":"339","article-title":"Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations","volume":"58","author":"Foia\u015f, C.","year":"1979","journal-title":"J. Math. Pures Appl. (9)","ISSN":"https:\/\/id.crossref.org\/issn\/0021-7824","issn-type":"print"},{"key":"12","isbn-type":"print","first-page":"413","article-title":"Incompressible fluid dynamics: some fundamental formulation issues","author":"Gresho, P. M.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0824307232"},{"key":"13","doi-asserted-by":"crossref","unstructured":"P.M. Gresho and R.L. Sani, On pressure boundary-conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids 7(10) (1987), 1111\u20131145.","DOI":"10.1002\/fld.1650071008"},{"key":"14","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0089647","volume-title":"Geometric theory of semilinear parabolic equations","volume":"840","author":"Henry, Daniel","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/3540105573"},{"key":"15","isbn-type":"print","first-page":"223","article-title":"Auxiliary flux and pressure conditions for Navier-Stokes problems","author":"Heywood, John G.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/3540097341"},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1137\/0719018","article-title":"Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization","volume":"19","author":"Heywood, John G.","year":"1982","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"17","first-page":"367","article-title":"Numerical approximation of two-dimensional convection-diffusion equations with multiple boundary layers","volume":"2","author":"Jung, Chang-Yeol","year":"2005","journal-title":"Int. J. Numer. Anal. Model.","ISSN":"https:\/\/id.crossref.org\/issn\/1705-5105","issn-type":"print"},{"key":"18","series-title":"Translations of Mathematical Monographs, Vol. 23","doi-asserted-by":"crossref","DOI":"10.1090\/mmono\/023","volume-title":"Linear and quasilinear equations of parabolic type","author":"Lady\u017eenskaja, O. A.","year":"1968"},{"key":"19","first-page":"481","article-title":"On the convergence of Fourier series defining a solution of a mixed problem for hyperbolic equations","volume":"85","author":"Lady\u017eenskaya, O.","year":"1952","journal-title":"Doklady Akad. Nauk SSSR (N.S.)"},{"key":"20","first-page":"395","article-title":"On solvability of the fundamental boundary problems for equations of parabolic and hyperbolic type","volume":"97","author":"Lady\u017eenskaya, O. A.","year":"1954","journal-title":"Dokl. Akad. Nauk SSSR (N.S.)"},{"key":"21","volume-title":"Quelques m\\'{e}thodes de r\\'{e}solution des probl\\`emes aux limites non lin\\'{e}aires","author":"Lions, J.-L.","year":"1969"},{"key":"22","isbn-type":"print","doi-asserted-by":"publisher","first-page":"508","DOI":"10.1007\/BFb0103275","article-title":"Semigroups of operators in Banach spaces","author":"Pazy, A.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/3540126864"},{"key":"23","doi-asserted-by":"publisher","first-page":"303","DOI":"10.2307\/1996861","article-title":"Differentiability of solutions to hyperbolic initial-boundary value problems","volume":"189","author":"Rauch, Jeffrey B.","year":"1974","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"issue":"1","key":"24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF02566671","article-title":"Smooth solutions of the heat and wave equations","volume":"55","author":"Smale, Stephen","year":"1980","journal-title":"Comment. Math. Helv.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-2571","issn-type":"print"},{"issue":"1","key":"25","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/0022-0396(82)90075-4","article-title":"Behaviour at time \ud835\udc61=0 of the solutions of semilinear evolution equations","volume":"43","author":"Temam, R.","year":"1982","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"key":"26","unstructured":"\\bysame, Navier-Stokes equations, AMS Chelsea Publishing, Providence, RI, 2001, Theory and numerical analysis, Reprint of the 1984 edition."},{"key":"27","unstructured":"Kevin E. Trenberth, Climate system modeling, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, Press Syndicate of the University of Cambridge, New York, NY, USA, 1992."},{"key":"28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2005811","article-title":"Convergence rates of parabolic difference schemes for non-smooth data","volume":"28","author":"Thom\u00e9e, Vidar","year":"1974","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02469-5\/S0025-5718-2011-02469-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02469-5\/S0025-5718-2011-02469-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:57:22Z","timestamp":1776790642000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02469-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,4,14]]},"references-count":28,"journal-issue":{"issue":"276","published-print":{"date-parts":[[2011,10]]}},"alternative-id":["S0025-5718-2011-02469-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02469-5","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":[[2011,4,14]]}}}