{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:17:44Z","timestamp":1787329064337,"version":"build-2736575974"},"reference-count":10,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We present a new modified nodal cubic spline collocation scheme for solving the Dirichlet problem for Poisson's equation on the unit square. We prove existence and uniqueness of a solution of the scheme and show how the solution can be computed on an $(N+1) \\times (N+1)$ uniform partition of the square with cost $O(N^2\\mbox{log} N)$ using a direct fast Fourier transform method. Using two comparison functions, we derive an optimal fourth order error bound in the continuous maximum norm. We compare our scheme with other modified nodal cubic spline collocation schemes; in particular, the one proposed by Houstis, Vavalis, and Rice in [SIAM J. Numer. Anal., 25 (1988), pp. 54\u201374]. We believe that our paper gives the first correct convergence analysis of a modified nodal cubic spline collocation for solving partial differential equations.<\/jats:p>","DOI":"10.1137\/050629033","type":"journal-article","created":{"date-parts":[[2008,1,25]],"date-time":"2008-01-25T18:02:54Z","timestamp":1201284174000},"page":"397-418","source":"Crossref","is-referenced-by-count":20,"title":["Modified Nodal Cubic Spline Collocation For Poisson's Equation"],"prefix":"10.1137","volume":"46","author":[{"given":"Abeer Ali","family":"Abushama","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bernard","family":"Bialecki","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,1,25]]},"reference":[{"key":"R1","unstructured":"J. H. Ahlberg, E. N. Nilson, and J. L. Walsh,\n                      The Theory of Splines and Their Applications\n                      , Academic Press, New York, 1967."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0714042"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S106482750139964X"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/040609793"},{"key":"R5","unstructured":"W. Cheney and D. Kincaid,\n                      Numerical Mathematics and Computing\n                      , Brooks\/Cole Publishing Co., Pacific Grove, CA, 1999."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/16.2.161"},{"key":"R7","unstructured":"C. de Boor,\n                      The Method of Projections as Applied to the Numerical Solution of Two Point Boundary Value Problems Using Cubic Splines\n                      , Ph.D. thesis, University of Michigan, Ann Arbor, MI, 1966."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/0725006"},{"key":"R9","unstructured":"A. A. Samarski,\n                      The Theory of Difference Schemes\n                      , Marcel Dekker, Inc., New York, 2001."},{"key":"R10","doi-asserted-by":"crossref","unstructured":"C. Van Loan,\n                      Computational Frameworks for the Fast Fourier Transform\n                      , Frontiers Appl. Math. 10, SIAM, Philadelphia, 1992.","DOI":"10.1137\/1.9781611970999"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/050629033","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:09:58Z","timestamp":1787324998000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/050629033"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/050629033"],"URL":"https:\/\/doi.org\/10.1137\/050629033","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}