{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T07:35:03Z","timestamp":1787384103930,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2000,1]]},"abstract":"<jats:p>The mixed-hybrid finite element discretization of Darcy's law and continuity equation describing the potential fluid flow problem in porous media leads to a symmetric indefinite linear system for the pressure and velocity vector components. As a method of solution the reduction to three Schur complement systems based on successive block elimination is considered. The first and second Schur complement matrices are formed eliminating the velocity and pressure variables, respectively, and the third Schur complement matrix is obtained by elimination of a part of Lagrange multipliers that come from the hybridization of a mixed method. The structural properties of these consecutive Schur complement matrices in terms of the discretization parameters are studied in detail. Based on these results the computational complexity of a direct solution method is estimated and compared to the computational cost of the iterative conjugategradient method applied to Schur complement systems. It is shown that due to special block structure the spectral properties of successive Schur complement matrices do not deteriorate and the approach based on the block elimination and subsequent iterative solution is well justified. Theoretical results are illustrated by numerical experiments.<\/jats:p>","DOI":"10.1137\/s1064827598339608","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"704-723","source":"Crossref","is-referenced-by-count":33,"title":["Schur Complement Systems in the Mixed-Hybrid Finite Element Approximation of the Potential Fluid Flow Problem"],"prefix":"10.1137","volume":"22","author":[{"given":"J.","family":"Maryska","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Rozlozn\u00edk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Tuma","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 User\u2019s Guide\n                      , SIAM, Philadelphia, 1992."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479898338147"},{"key":"R3","unstructured":"J. Bear,\n                      Dynamics of Fluids in Porous Media\n                      , Elsevier, New York, 1972."},{"key":"R4","unstructured":"R. Hoppe, B. Wohlmuth, Adaptive mixed hybrid and macro\u2010hybrid finite element methods, Proceedings of the Algoritmy\u201897 Conference on Scientific Computing (Zuberec), Vol. 67, 1998, 159\u20131792000a:65161"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479893246765"},{"key":"R6","unstructured":"H. C. Elman,\n                      Iterative Methods for Large, Sparse, Nonsymmetric Systems of Linear Equations\n                      , Research Report 229, Yale University, New Haven, CT, 1982."},{"key":"R7","unstructured":"J. R. Gilbert, G. L. Miller, and S. H. Teng,\n                      Geometric mesh partitioning: Implementation and experiments\n                      , in Proceedings 9th International Parallel Processing Symposium, Santa Barbara, CA, IEEE Computer Society Press, Los Alamitos, CA, 1995, pp. 418\u2013427."},{"key":"R8","unstructured":"G. H. Golub and C. F. van Loan,\n                      Matrix Computations\n                      , 2nd ed., The Johns Hopkins University Press, Baltimore, 1989."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.6028\/jres.049.044"},{"key":"R10","volume-title":"Accuracy and stability of numerical algorithms","author":"Higham Nicholas","year":"1996"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1002\/num.1690080302"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/0716027"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-1506(199805\/06)5:3<219::AID-NLA134>3.0.CO;2-7"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(95)00066-6"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-1506(199611\/12)3:6<525::AID-NLA94>3.0.CO;2-X"},{"key":"R16","volume-title":"Sparse factorization with applications to linear least squares problems","author":"Matstoms Pontus","year":"1994"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596303491"},{"key":"R18","unstructured":"Gary Miller, Shang\u2010Hua Teng, William Thurston, Stephen Vavasis, Automatic mesh partitioning, IMA Vol. Math. Appl., Vol. 56, Springer, New York, 1993, 57\u20138496c:68157"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827594262613"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(81)90232-9"},{"key":"R21","unstructured":"I. Perugia and V. Simoncini,\n                      Block\u2010diagonal and indefinite symmetric preconditioners for mixed finite element formulations\n                      , Technical report 1159, IAN CNR, Pavia, Italy, Numer. Linear Algebra Appl., to appear."},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/0613054"},{"key":"R23","unstructured":"J. Varah, A survey of iterative methods for sparse linear systems, Proceedings of the fifteenth Manitoba conference on numerical mathematics and computing (Winnipeg, Man., 1985), Vol. 51, 1986, 83\u20139288h:65080"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1002\/nla.1680010405"},{"key":"R25","unstructured":"Daniel Spielman, Shang\u2010Hua Teng, Spectral partitioning works: planar graphs and finite element meshes, IEEE Comput. Soc. Press, Los Alamitos, CA, 1996, 96\u201310598d:65144"},{"key":"R26","unstructured":"H. van der Vorst,\n                      Parallel Iterative Solution Methods for Linear Systems Arising from Discretized PDE\u2019s\n                      , Workshop Lecture Notes, in Special Course on Parallel Computing in CFD, AGARD\u2010R\u2010807, AGARD, Neuilly\u2010sur\u2010Seine, France, 1995;"},{"key":"R26","unstructured":"also available online at http:\/\/www.math.ruu.nl.people\/vorst\/agard.ps.gz."},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050251"},{"key":"R28","unstructured":"R\u00fcdiger Weiss, Parameter\u2010free iterative linear solvers, Mathematical Research, Vol. 97, Akademie Verlag, 1996, 217\u2013097m:65074"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827598339608","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:40:04Z","timestamp":1787334004000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827598339608"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,1]]},"references-count":29,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2000,1]]}},"alternative-id":["10.1137\/S1064827598339608"],"URL":"https:\/\/doi.org\/10.1137\/s1064827598339608","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,1]]}}}