{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:28:44Z","timestamp":1787336924627,"version":"build-2736575974"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[1997,3]]},"abstract":"<jats:p>In this paper a multilevel algorithm for the solution of the cell vertex finite volume Cauchy--Riemann equations is developed. These equations provide a linear algebraic system obtained by the finite volume cell vertex discretization of the inhomogeneous Cauchy--Riemann equations. Both square and triangular cells are employed. The system of linear equations resulting from the cell vertex discretization is overdetermined and its solution is considered in the least squares sense. By this approach a consistent algebraic problem is obtained which differs from the original one by O(h2) perturbation of the right-hand side. A suitable cell-based convergent smoothing iteration is presented which is naturally linked to the least squares formulation. Hence a standard multilevel scheme is presented and discussed which combines the given smoother and a cell-based transfer operator of the residuals and a node-based prolongation operator of the unknown variables. Some remarkable reduction properties of these operators are shown. A full multilevel algorithm is constructed which solves the discrete problem to the level of truncation error by employing one multilevel cycle at each current level of discretization.<\/jats:p>","DOI":"10.1137\/s1064827595281952","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"441-459","source":"Crossref","is-referenced-by-count":6,"title":["Multilevel Solution of Cell Vertex Cauchy--Riemann Equations"],"prefix":"10.1137","volume":"18","author":[{"given":"A.","family":"Borz\u00ec","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K. W.","family":"Morton","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"E.","family":"S\u00fcli","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Vanmaele","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","unstructured":"A. Brandt,\n                      Multi\u2010grid techniques: 1984 guide with applications to fluid dynamics\n                      , GMD\u2010Studien no. 85, St. Augustin, Germany, 1984."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(86)90095-0"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"A. Brandt and N. Dinar,\n                      Multigrid solutions to elliptic flow problems, Numerical methods for partial differential equations\n                      , ed. S. V. Parter, Academic Press, New York, 1979.","DOI":"10.1016\/B978-0-12-546050-7.50008-3"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(92)90049-5"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/0914039"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/0727069"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1017\/9781139171984"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1993.1194"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/0722016"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(82)90032-8"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1979-0521278-7"},{"key":"R12","unstructured":"M. Giles,\n                      Accuracy of node\u2010based solutions on irregular meshes\n                      ,\n                      Proceedings of the 11th International Conference on Numerical Methods in Fluid Dynamics\n                      , D. L. Dwoyer, M. Y. Hussaini and R. G. Voigt, eds., Vol. 323 of\n                      Lecture Notes in Physics\n                      , Springer\u2010Verlag, Berlin, 1988."},{"key":"R13","doi-asserted-by":"crossref","unstructured":"W. Hackbusch,\n                      Multi\u2010Grid Methods and Applications\n                      , Springer\u2010Verlag, Heidelberg, 1985.","DOI":"10.1007\/978-3-662-02427-0"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1982-0645657-2"},{"key":"R15","unstructured":"D. Lindquist and M. Giles,\n                      A comparison of numerical schemes on triangular and quadrilateral meshes\n                      ,\n                      Proceedings of the 11th International Conference on Numerical Methods in Fluid Dynamics\n                      , D. L. Dwoyer, M. Y. Hussaini and R. G. Voigt, eds., Vol. 323 of\n                      Lecture Notes in Physics\n                      , Springer\u2010Verlag, Berlin, 1988."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(74)90069-2"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1993-1153168-0"},{"key":"R18","unstructured":"K. W. Morton,\n                      Finite volume and finite element methods for the steady Euler equations of gas dynamics\n                      ,\n                      The Mathematics of Finite Elements and Applications VI MAFELAP\n                      , ed. J. R. Whiteman, Academic Press, New York\/London, 1988."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(89)90094-6"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1994.1161"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/0718023"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"J. W. Ruge and K. St\u00fcben,\n                      Algebraic multigrid\n                      ,\n                      Multigrid Methods\n                      , ed. S. F. McCormick, SIAM, Philadelphia, 1987.","DOI":"10.1137\/1.9781611971057.ch4"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1992-1134740-X"},{"key":"R24","unstructured":"S. Ta\u2019asan,\n                      Canonical forms of multidimensional steady inviscid flows\n                      , Institute for Computer Applications in Science and Engineering, ICASE 93\u201034, Hampton, Virginia, 1993."},{"key":"R25","unstructured":"S. Ta\u2019asan,\n                      Canonical\u2010variables multigrid method for steady\u2010state Euler equations\n                      , Institute for Computer Applications in Science and Engineering, ICASE 94\u201014, Hampton, Virginia, 1994."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994276384"},{"key":"R27","unstructured":"D. M. Young and R. T. Gregory,\n                      A Survey of Numerical Mathematics, Vol. II\n                      , Addison\u2010Wesley Publishing Company, 1973."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827595281952","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:36:53Z","timestamp":1787333813000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827595281952"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,3]]},"references-count":27,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1997,3]]}},"alternative-id":["10.1137\/S1064827595281952"],"URL":"https:\/\/doi.org\/10.1137\/s1064827595281952","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[1997,3]]}}}