{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:33:45Z","timestamp":1787337225166,"version":"build-2736575974"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2000,1]]},"abstract":"<jats:p>This paper describes a parallel convection-diffusion solver which may be used as part of a Navier--Stokes solver for three-dimensional channel flow at moderately large Reynolds numbers [S. Turek, Efficient Solvers for Incompessible Flow Problems: An Algorithmic Approach in View of Computational Aspects, Springer-Verlag, 1999]. The solver uses a multiplicative Schwarz domain decomposition with overlapping subdomains to solve singularly perturbed convection-diffusion equations where convection is dominant. Upwind finite differences are used for the spatial discretization. The algorithm uses special features of the singularly perturbed convection-diffusion operator. The error due to a local perturbation of the boundary conditions decays extremely fast, in the upwind as well as the crosswind direction, so the overlap in the domain decomposition can be kept to a minimum. The algorithm parallelizes well and is particularly suited for applications in three dimensions. Results of two- and three-dimensional numerical experiments are presented.<\/jats:p>","DOI":"10.1137\/s1064827598335854","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"891-916","source":"Crossref","is-referenced-by-count":22,"title":["A Parallel Schwarz Method for a Convection-Diffusion Problem"],"prefix":"10.1137","volume":"22","author":[{"given":"M.","family":"Garbey","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu. A.","family":"Kuznetsov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu. V.","family":"Vassilevski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02238647"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"Q. Dihn, R. Glowinski, J. P\u00e9riaux, Solving elliptic problems by domain decomposition methods with applications, Academic Press, Orlando, FL, 1984, 395\u2013426764246","DOI":"10.1016\/B978-0-12-100560-3.50032-9"},{"key":"R3","volume-title":"Asymptotic analysis of singular perturbations","author":"Eckhaus Wiktor","year":"1979"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827593258437"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142995285587"},{"key":"R6","unstructured":"M. Garbey and Yu. A. Kuznetsov,\n                      Parallel Schwarz Algorithm for Equations with Singular\u2010Perturbed Convection\u2010Diffusion Operator\n                      , Preprint 238, C. N. R. S. U. M. R. 5585, Universit\u00e9 Claude Bernard Lyon 1, 1996."},{"key":"R7","unstructured":"M. Garbey, Yu. A. Kuznetsov, and Yu. V. Vassilevski,\n                      Parallel Schwarz Algorithm for a Problem with a Singular\u2010Perturbed Convection\u2010Diffusion Operator\n                      , Preprint CDCSP\u201097\u201008, CDCSP, Universit\u00e9 Claude Bernard Lyon 1, 1997; also available online from http:\/\/www.univ\u2010lyon1.fr\/cdcsp."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(97)82364-6"},{"key":"R9","unstructured":"F. K. Hebeker and Yu. A. Kuznetsov,\n                      Unsteady Convection and Convection\u2010Diffusion Problems via Overlapping Domain\u2010Decomposition Methods\n                      , Preprint 93\u201054 (SFB 359), University of Heidelberg, Germany, 1993."},{"key":"R10","unstructured":"Y. Iliash, Y. Kuznetsov, and Y. Vassilevski,\n                      Efficient Parallel Solvers for Two Dimensional Potential Flow and Convection\u2010Diffusion Problems on Nonmatching Grids\n                      , Report 357, Universit\u00e4t Augsburg, Germany, 1996."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1987-0890252-8"},{"key":"R12","unstructured":"Yu. Kuznetsov, Novye algoritmy priblizhennoi realizatsii neyavnykh raznostnykh skhem, Y\u00b8r Preprint [Preprint], Vol. 142, Akad. Nauk SSSR Otdel Vychisl. Mat., 1987, 24\u20130, Translated in Soviet J. Numer. Anal. Math. Modelling\n                      3\n                      (1988), no. 2, 99\u201311489f:65114"},{"key":"R13","unstructured":"Yu. Kuznetsov, Domain decomposition methods for unsteady convection\u2010diffusion problems, SIAM, Philadelphia, PA, 1990, 211\u20132271102024"},{"key":"R14","unstructured":"Yu. Kuznetsov, Overlapping domain decomposition methods for FE\u2010problems with elliptic singular perturbed operators, SIAM, Philadelphia, PA, 1991, 223\u201324192c:65145"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01405198"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"J. M. Ortega,\n                      Numerical Analysis: A Second Course\n                      , SIAM, Philadelphia, 1990.","DOI":"10.1137\/1.9781611971323"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596302825"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596305829"},{"key":"R19","first-page":"151","volume":"2","author":"Rannacher M.","year":"1994","journal-title":"East\u2010West J. Numer. 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