{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:34:00Z","timestamp":1787337240032,"version":"build-2736575974"},"reference-count":8,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2001,1]]},"abstract":"<jats:p>\n                    A Legendre spectral Galerkin method is presented for the solution of the biharmonic Dirichlet problem on a square. The solution and its Laplacian are approximated using the set of basis functions suggested by Shen, which are linear combinations of two Legendre polynomials. A Schur complement approach is used to reduce the resulting linear system to one involving the approximation of the Laplacian of the solution on the two vertical sides of the square. The Schur complement system is solved by a preconditioned conjugate gradient method or the Cholesky method. The total cost of the algorithm is O(N\n                    <jats:sup>3<\/jats:sup>\n                    ). Numerical results demonstrate the spectral convergence of the method.\n                  <\/jats:p>","DOI":"10.1137\/s1064827598342407","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"1549-1569","source":"Crossref","is-referenced-by-count":38,"title":["A Legendre Spectral Galerkin Method for the Biharmonic Dirichlet Problem"],"prefix":"10.1137","volume":"22","author":[{"given":"Bernard","family":"Bialecki","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andreas","family":"Karageorghis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","unstructured":"E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen,\n                      LAPACK Users\u2019 Guide\n                      , SIAM, Philadelphia, 1992."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2000160"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596298233"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet and P.\u2010A. Raviart,\n                      A mixed finite element method for the biharmonic equation\n                      , in Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, ed., Academic Press, New York, 1974, pp. 125\u2013143.","DOI":"10.1016\/B978-0-12-208350-1.50009-1"},{"key":"R5","unstructured":"D. B. Knudson,\n                      A Piecewise Hermite Bicubic Finite Element Galerkin Method for the Biharmonic Dirichlet Problem\n                      , Ph.D. thesis, Colorado School of Mines, Golden, CO, 1997."},{"key":"R6","unstructured":"J. Shen,\n                      Efficient Chebyshev\u2010Legendre Galerkin methods for elliptic problems\n                      , in Proceedings of the Third International Conference on Spectral and High Order Methods, A. V. Ilin and L. R. Scott, eds., Houston J. Math., Houston, TX, 1996, pp. 233\u2013239."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0915089"},{"key":"R8","unstructured":"B. P. Tj\u00f8stheim,\n                      private communication\n                      ."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827598342407","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:51:42Z","timestamp":1787334702000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827598342407"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,1]]},"references-count":8,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2001,1]]}},"alternative-id":["10.1137\/S1064827598342407"],"URL":"https:\/\/doi.org\/10.1137\/s1064827598342407","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,1]]}}}