{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:16:09Z","timestamp":1787328969400,"version":"build-2736575974"},"reference-count":31,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>We present a finite difference scheme, applicable to general irregular planar domains, to approximate the biharmonic equation. The irregular domain is embedded in a Cartesian grid. In order to approximate $\\Delta^{2}\\Phi$ at a grid point we interpolate the data on the (irregular) stencil by a polynomial of degree six. The finite difference scheme is $\\Delta^{2}Q_{\\Phi}(0,0)$, where $Q_{\\Phi}$ is the interpolation polynomial. The interpolation polynomial is not uniquely determined. We present a method to construct such an interpolation polynomial and prove that our construction is second order accurate. For a regular stencil, [M. Ben-Artzi, J.-P. Croisille, and D. Fishelov, SIAM J. Sci. Comput., 31 (2008), pp. 303\u2013333] shows that the proposed interpolation polynomial is fourth order accurate. We present some suitable numerical examples.<\/jats:p>","DOI":"10.1137\/080718784","type":"journal-article","created":{"date-parts":[[2009,9,28]],"date-time":"2009-09-28T15:18:51Z","timestamp":1254151131000},"page":"3087-3108","source":"Crossref","is-referenced-by-count":35,"title":["A Compact Difference Scheme for the Biharmonic Equation in Planar Irregular Domains"],"prefix":"10.1137","volume":"47","author":[{"given":"M.","family":"Ben-Artzi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"I.","family":"Chorev","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J.-P.","family":"Croisille","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D.","family":"Fishelov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,9,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S1464827596296970"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2426(199707)13:4<375::AID-NUM5>3.0.CO;2-I"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/1985190100071"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9991(03)00342-5"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.11.024"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/05062915X"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/070694168"},{"key":"R8","unstructured":"M. Ben-Artzi, I. Chorev, J.P. Croisille, and D. Fishelov,\n                      The biharmonic operator in irregular planar domains: Convergence of a compact scheme\n                      , in preparation, 2009."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/0720004"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0726062"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595292710"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-007-9043-6"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF00252458"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"P. A. Ciarlet and P. G. Raviart,\n                      A mixed finite element method for the biharmonic equation\n                      , in Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, New York, 1974, pp. 125\u2013145.","DOI":"10.1016\/B978-0-12-208350-1.50009-1"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142999359773"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(91)90003-O"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/0712058"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2006.11.022"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"A. Greenbaum, L. Greengard, and A. Mayo,\n                      On the numerical solution of the biharmonic equation in the plane\n                      , Phys. 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Fluid Mech. 11, Vieweg, Braunschweig, 1985.","DOI":"10.1007\/978-3-663-14245-4_7"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(99)00321-0"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/0721021"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-008-9186-8"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"L. L. Schumaker,\n                      Spline Functions: Basic Theory\n                      , 3rd ed., Cambridge University Press, Cambridge, UK, 2007.","DOI":"10.1017\/CBO9780511618994"},{"key":"R29","unstructured":"B. Schiff, D. Fishelov, and J. R. Whiteman,\n                      Determination of a stress intensity factor using local mesh refinement\n                      , in Mathematics of Finite Elements and Applications, Academic Press, London-New York, 1979, pp. 55\u201364."},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.10.014"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(84)90015-9"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080718784","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:08:12Z","timestamp":1787324892000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080718784"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":31,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/080718784"],"URL":"https:\/\/doi.org\/10.1137\/080718784","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}