{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:40:05Z","timestamp":1787330405997,"version":"build-2736575974"},"reference-count":32,"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":[[2007,1]]},"abstract":"<jats:p>Although the asymptotic complexity of direct methods for the solution of large sparse finite element systems arising from second-order elliptic partial differential operators is far from being optimal, these methods are often preferred over modern iterative methods. This is mainly due to their robustness. In this article it is shown that an approximate $LU$ decomposition exists which can be computed in the algebra of hierarchical matrices with almost linear complexity and with the same robustness as the classical $LU$ decomposition. Low-precision approximations may be used for preconditioning iterative solvers. As a byproduct we prove that Schur complements of stiffness matrices can be approximated with almost linear complexity.<\/jats:p>","DOI":"10.1137\/060669747","type":"journal-article","created":{"date-parts":[[2007,7,25]],"date-time":"2007-07-25T18:00:19Z","timestamp":1185386419000},"page":"1472-1494","source":"Crossref","is-referenced-by-count":49,"title":["Why Finite Element Discretizations Can Be Factored by Triangular Hierarchical Matrices"],"prefix":"10.1137","volume":"45","author":[{"given":"Mario","family":"Bebendorf","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899358194"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-04-01716-8"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-004-0099-6"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479803437621"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/PL00005410"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-002-1469-6"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-002-0445-6"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899353373"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"E. Cuthill and J. McKee,\n                      Reducing the bandwidth of sparse symmetric matrices\n                      , in Proceedings of the 24th National Conference of the ACM, 1969, pp. 157\u2013172.","DOI":"10.1145\/800195.805928"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1002\/nla.1680020208"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(97)00008-4"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(98)00069-8"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/0710032"},{"key":"R14","unstructured":"A. George and J. Liu,\n                      Computer Solution of Large Sparse Positive Definite Systems\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1981."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/1031001"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-003-0019-1"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/s006070050015"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/PL00021408"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396324"},{"key":"R20","unstructured":"T. J. R. Hughes and A. Brooks,\n                      A multidimensional upwind scheme with no crosswind diffusion\n                      , in Finite Element Methods for Convection Dominated Flows, AMD 34, T. J .R. Hughes, ed., ASME, New York, 1979, pp. 19\u201335."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1994.1116"},{"key":"R22","unstructured":"M. Lintner,\n                      L\u00f6sung der\n                      2\n                      D Wellengleichung mittels hierarchischer Matrizen\n                      , Doctoral thesis, Technische Universit\u00e4t M\u00fcnchen, Munich, Germany, 2002."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827594262613"},{"key":"R24","unstructured":"S. Rjasanow, I. Ibraghimov, and K. Straube,\n                      Hierarchical Cholesky decomposition of sparse matrices arising from curl-curl-equations\n                      , J. Numer. Math., to appear."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(85)90002-6"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"D. J. Rose,\n                      A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations\n                      , in Graph Theory and Computing, Academic Press, New York, 1972, pp. 183\u2013217.","DOI":"10.1016\/B978-1-4832-3187-7.50018-0"},{"key":"R27","unstructured":"Y. Saad,\n                      Iterative Methods for Sparse Linear Systems\n                      , PWS Publishing, Boston, 1996."},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1007\/s007910050004"},{"key":"R29","unstructured":"B. F. Smith, P. E. Bj\u00f8rstad, and W. D. Gropp,\n                      Domain Decomposition: Parallel Multi-level Methods for Elliptic Partial Differential Equations\n                      , Cambridge University Press, Cambridge, UK, 1996."},{"key":"R30","doi-asserted-by":"crossref","unstructured":"P. Starr and V. Rokhlin,\n                      On the numerical solution of two-point boundary value problems.\n                      II, Comm. Pure Appl. Math., 47 (1994), pp. 1117\u20131159.","DOI":"10.1002\/cpa.3160470806"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1007\/BF02575706"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2003.09.040"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060669747","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:03:22Z","timestamp":1787328202000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060669747"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1]]},"references-count":32,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["10.1137\/060669747"],"URL":"https:\/\/doi.org\/10.1137\/060669747","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,1]]}}}