{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:25:23Z","timestamp":1787336723035,"version":"build-2736575974"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[1997,11]]},"abstract":"<jats:p>A class of algebraic multi-p methods for solving the p-version of the finite element equations is first presented and discussed. It is then shown howthese multi-p methods can be used as preconditioners for the conjugate gradient (CG) method. In particular, it is shown that given any preconditioner Mp to CG, a multi-p preconditioner Bp based on Mp can be constructed, which leads to a smaller condition number (and hence faster convergence).<\/jats:p>\n                  <jats:p>Numerical experiments on representative problems indicate that the condition numbers after multi-p preconditionings are, in fact, independent of p. The numerical results also show greater efficiency for the preconditioned conjugate gradient (PCG) method with the multi-p preconditioners in terms of number of iterations and CPU time when compared with two sophisticated linear equation solvers: (1) a direct frontal solver specially designed for the p-version of the finite element analysis; (2) a highly tuned PCG code in ITPACK2C, http:\/\/www.netlib.org.itpack\/, 1994. Preliminary comparisons of the number of iterations are also made with ROCKITS [Solvers International, Inc., Boulder, CO, 1994], a new commercial code used for the p-version.<\/jats:p>","DOI":"10.1137\/s1064827595279368","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"1676-1697","source":"Crossref","is-referenced-by-count":12,"title":["Multi-p Preconditioners"],"prefix":"10.1137","volume":"18","author":[{"given":"Ning","family":"Hu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xian-zhong","family":"Guo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"I. Norman","family":"Katz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0718033"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0724049"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(90)90011-A"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0728034"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620280813"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(89)90365-8"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(90)90036-L"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1986-0842125-3"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1989-0970699-3"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/1031003"},{"key":"R11","unstructured":"G. Golub and C. Van Loan,\n                      Matrix Computations\n                      , 2nd ed., The Johns Hopkins University Press, Baltimore, MD, 1989."},{"key":"R12","unstructured":"L. Khei\u02d8geman, D. Yang, Prikladnye iteratsionnye metody, \u201cMir\u201d, 1986, 448\u20130, Translated from the English by A. Yu. Er\u00ebmin and I. E. Kaporin; Edited and with a preface by Yu. A. Kuznetsov88b:65048"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.6028\/jres.049.044"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/0916076"},{"key":"R15","unstructured":"C. Johnson,\n                      Numerical Solution of Partial Differential Equations by the Finite Element Method\n                      , Cambridge University Press, Cambridge, UK, 1990."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(91)90398-G"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/BF01932747"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994265176"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620290513"},{"key":"R20","unstructured":"J. Mandel,\n                      Hierarchial preconditioning and partial orthogonalization for the p\u2010version finite element method\n                      , in Proc. 1989 Conference on Domain Decomposition Methods, Houston, TX, T. Chan, R. Glowinski, and O. Widlund, eds., SIAM, Philadelphia, PA, 1990."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(90)90017-G"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385611"},{"key":"R23","unstructured":"J. Mandel,\n                      Adaptive iterative solvers in finite elements\n                      , in Solving Large Scale Problems in Mechanics: Development and Application of Computational Solution Methods, M. Papadrakakis, ed., John Wiley, Chichester, UK, 1992."},{"key":"R24","unstructured":"J. Mandel, S. McCormick, R. Bank, Variational multigrid theory, Frontiers Appl. Math., Vol. 3, SIAM, Philadelphia, PA, 1987, 131\u2013177972757"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/0719036"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/0721018"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1137\/0721045"},{"key":"R28","unstructured":"J. F. Maitre and F. Musy,\n                      Algebraic formalization of the multigrid method in the symmetric and positive definite case\u2014a convergence estimation for the V\u2010cycle\n                      , in Multigrid Methods for Integral and Differential Equations, D. J. Paddon and H. Holstein, eds., Institute of Mathematics and Its Applications Conference Series 3, Clarendon Press, Oxford, UK, 1985, pp. 213\u2013224."},{"key":"R29","unstructured":"B. A. Szabo and I. Babuska,\n                      Finite element analysis\n                      , John Wiley, New York, 1991."},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1137\/1034116"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389538"},{"key":"R32","unstructured":"STRESS CHECK,\n                      User\u2019s Manual\n                      , Engineering Software Research and Development, Inc., St. Louis, MO, 1994;"},{"key":"R32","unstructured":"also available online from http:\/\/www.ESRD.com\/."},{"key":"R33","unstructured":"ITPACK2C, R. Kincaid, J. R. Respess, R. G. Grimes, 1994;"},{"key":"R33","unstructured":"also available online from http:\/\/www.netlib.org\/itpack\/."},{"key":"R34","unstructured":"ROCKITS, Solvers International, Inc., Boulder, CO, 1994."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827595279368","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:28:04Z","timestamp":1787333284000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827595279368"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,11]]},"references-count":36,"journal-issue":{"issue":"6","published-print":{"date-parts":[[1997,11]]}},"alternative-id":["10.1137\/S1064827595279368"],"URL":"https:\/\/doi.org\/10.1137\/s1064827595279368","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[1997,11]]}}}