{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T04:43:18Z","timestamp":1787373798226,"version":"build-2736575974"},"reference-count":21,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2007,1]]},"abstract":"<jats:p>This work presents techniques, theory, and numbers for multigrid in a general d-dimensional setting. The main focus of this paper is the multigrid convergence for high-dimensional partial differential equations on non-equidistant grids such as may be encountered in a sparse-grid solution. As a model problem we have chosen the anisotropic stationary diffusion equation on a rectangular hypercube. We present some techniques for building the general d-dimensional adaptations of the multigrid components and propose grid-coarsening strategies to handle anisotropies that are induced due to discretization on a non-equidistant grid. Apart from the practical formulas and techniques, we present\u2014in detail\u2014the smoothing analysis of the point $\\omega$-red-black Jacobi method for a general multidimensional case. We show how relaxation parameters may be evaluated efficiently and used for better convergence. This analysis incorporates full and partial doubling and quadrupling coarsening strategies as well as the second- and the fourth-order finite difference operators. Finally we present some results derived from numerical experiments based on the test problem.<\/jats:p>","DOI":"10.1137\/060665695","type":"journal-article","created":{"date-parts":[[2007,7,19]],"date-time":"2007-07-19T18:00:20Z","timestamp":1184868020000},"page":"1613-1636","source":"Crossref","is-referenced-by-count":21,"title":["Multigrid for High-Dimensional Elliptic Partial Differential Equations on Non-equidistant Grids"],"prefix":"10.1137","volume":"29","author":[{"given":"H. bin","family":"Zubair","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C. W.","family":"Oosterlee","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R.","family":"Wienands","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,7,18]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/040604959"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1977-0431719-X"},{"key":"R3","unstructured":"J. Elf, P. L\u00f6tstedt, and P. Sj\u00f6berg,\n                      Problems of high dimension in molecular biology\n                      , in Proceedings of the 19th GAMM-Seminar Leipzig, Max-Planck-Institute, Leipzig, 2003, pp. 21\u201330."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0207(19970315)40:5<887::AID-NME93>3.0.CO;2-I"},{"key":"R5","unstructured":"G. Horton and S. Vandewalle,\n                      A Space-Time Multigrid Method for Parabolic PDEs\n                      , Technical report 6\/93, IMMD 3, Universitaet Erlangen, 1993."},{"key":"R6","unstructured":"Y.K. Kwok,\n                      Mathematical Models of Financial Derivatives\n                      , 2nd ed., Springer Finance, Springer, Singapore, 1998."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2005.02.020"},{"key":"R8","unstructured":"C. Reisinger and G. Wittum,\n                      On multigrid for anisotropic equations and variational inequalities\n                      , Comput. Vis. Sci. (2004)."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/060649616"},{"key":"R10","unstructured":"H.W. Steeb,\n                      Kronecker Product of Matrices and Applications\n                      , Wissenschaftsverlag, 1991."},{"key":"R11","doi-asserted-by":"crossref","unstructured":"K. St\u00fcben and U. Trottenberg,\n                      Multigrid Methods: Fundamental Algorithms, Model Problem Analysis and Applications\n                      , Springer, Berlin, 1982.","DOI":"10.1007\/BFb0069928"},{"key":"R12","unstructured":"C. Reisinger and G. Wittum,\n                      On the Construction of Fast Solvers for Elliptic Equations\n                      , Technical report, von Karman Institute for Fluid Dynamics, Rhode-Saint-Genese, Belgium, 1982."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(86)90112-8"},{"key":"R14","unstructured":"U. Trottenberg, C.W. Oosterlee, and A. Sch\u00fcller,\n                      Multigrid\n                      , Academic Press, New York, 2001."},{"key":"R15","unstructured":"P. Wesseling,\n                      An Introduction to Multigrid Methods\n                      , Pure Appl. Math., John Wiley & Sons, New York, 1992."},{"key":"R16","unstructured":"R. Wienands,\n                      Extended Local Fourier Analysis for Multigrid: Optimal Smoothing, Coarse Grid Correction, and Preconditioning\n                      , Ph.D. thesis, University of Cologne, 2001."},{"key":"R17","doi-asserted-by":"crossref","unstructured":"R. Wienands and W. Joppich,\n                      Practical Fourier Analysis for Multigrid Methods\n                      , Numerical Insights, Chapman and Hall\/CRC, Boca Raton, FL, 2005.","DOI":"10.1201\/9781420034998"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/0732051"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/0917013"},{"key":"R20","unstructured":"Z. You-lan, W. Xiaonan, and C. I-Liang,\n                      Derivative Securities and Difference Methods\n                      , Springer Finance, Springer Science+Business Media Inc. USA, New York, 2004."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-005-0581-x"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060665695","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:58:25Z","timestamp":1787335105000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060665695"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["10.1137\/060665695"],"URL":"https:\/\/doi.org\/10.1137\/060665695","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,1]]}}}