{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:45:07Z","timestamp":1776728707063,"version":"3.51.2"},"reference-count":35,"publisher":"American Mathematical Society (AMS)","issue":"242","license":[{"start":{"date-parts":[[2003,11,4]],"date-time":"2003-11-04T00:00:00Z","timestamp":1067904000000},"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":"

\n In this paper, theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space-time grids. We consider parabolic equations containing a linear reaction term on a space-time domain\n \n \n \n \n \n \u03a9\n \n <\/mml:mi>\n \n \u00d7\n \n <\/mml:mo>\n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n T<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\Omega \\times [0,T]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n which is decomposed into an overlapping collection of cylindrical subregions of the form\n \n \n \n \n \n \n \u03a9\n \n <\/mml:mi>\n \n l<\/mml:mi>\n <\/mml:mrow>\n \n \n \u2217\n \n <\/mml:mo>\n <\/mml:mrow>\n <\/mml:msubsup>\n \n \u00d7\n \n <\/mml:mo>\n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n T<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\Omega _{l}^{\\ast } \\times [0,T]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , for\n \n \n \n \n l<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n l=1, \\dotsc , p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Each of the space-time domains\n \n \n \n \n \n \n \u03a9\n \n <\/mml:mi>\n \n l<\/mml:mi>\n <\/mml:mrow>\n \n \n \u2217\n \n <\/mml:mo>\n <\/mml:mrow>\n <\/mml:msubsup>\n \n \u00d7\n \n <\/mml:mo>\n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n T<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\Omega _{l}^{\\ast } \\times [0,T]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are assumed to be independently grided (in parallel) according to the local geometry and space-time regularity of the solution, yielding space-time grids with mesh parameters\n \n \n \n \n h<\/mml:mi>\n \n l<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n h_{l}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \u03c4\n \n <\/mml:mi>\n \n l<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \\tau _{l}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In particular, the different space-time grids need not match on the regions of overlap, and the time steps\n \n \n \n \n \n \u03c4\n \n <\/mml:mi>\n \n l<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \\tau _{l}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n can differ from one grid to the next. We discretize the parabolic equation on each local grid by employing an explicit or implicit\n \n \n \n \n \u03b8\n \n <\/mml:mi>\n \\theta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -scheme in time and a finite difference scheme in space satisfying a discrete maximum principle. The local discretizations are coupled together, without the use of Lagrange multipliers, by requiring the boundary values on each space-time grid to match a suitable interpolation of the solution on adjacent grids. The resulting global discretization yields a large system of coupled equations which can be solved by a parallel Schwarz iterative procedure requiring some communication between adjacent subregions. Our analysis employs a contraction mapping argument. Applications of the results are briefly indicated for reaction-diffusion equations with contractive terms and\n heterogeneous<\/italic>\n hyperbolic-parabolic approximations of parabolic equations.\n <\/p>","DOI":"10.1090\/s0025-5718-02-01462-x","type":"journal-article","created":{"date-parts":[[2003,2,10]],"date-time":"2003-02-10T11:10:52Z","timestamp":1044875452000},"page":"619-656","source":"Crossref","is-referenced-by-count":7,"title":["Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids"],"prefix":"10.1090","volume":"72","author":[{"given":"T.","family":"Mathew","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"G.","family":"Russo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2002,11,4]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1090\/conm\/218\/02999","article-title":"Nonmatching grids for fluids","author":"Achdou, Yves","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0821809881"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"551","DOI":"10.1137\/S0036142997321005","article-title":"Iterative substructuring preconditioners for mortar element methods in two dimensions","volume":"36","author":"Achdou, Yves","year":"1999","journal-title":"SIAM J. 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