{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T15:59:10Z","timestamp":1774886350954,"version":"3.50.1"},"reference-count":54,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2015,1,14]],"date-time":"2015-01-14T00:00:00Z","timestamp":1421193600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2015,11]]},"DOI":"10.1007\/s00211-015-0697-6","type":"journal-article","created":{"date-parts":[[2015,1,13]],"date-time":"2015-01-13T08:41:52Z","timestamp":1421138512000},"page":"517-540","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A comparison of the Extrapolated Successive Overrelaxation and the Preconditioned Simultaneous Displacement methods for augmented linear systems"],"prefix":"10.1007","volume":"131","author":[{"given":"M. A.","family":"Louka","sequence":"first","affiliation":[]},{"given":"N. M.","family":"Missirlis","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2015,1,14]]},"reference":[{"key":"697_CR1","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s10596-008-9121-y","volume":"13","author":"T Arbogast","year":"2009","unstructured":"Arbogast, T., Gomez, M.S.M.: A discretization and multigrid solver for a Darcy\u2013Stokes system of three dimentional vuggy porous media. Comput. Geosci. 13, 331\u2013348 (2009)","journal-title":"Comput. Geosci."},{"key":"697_CR2","unstructured":"Arrow, K., Hurwicz, L., Uzawa, H.: Studies in Nonlinear Programming. Stanford University Press, Stanford, CA (1958)"},{"issue":"6","key":"697_CR3","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1016\/j.apnum.2011.07.010","volume":"62","author":"C Bacuta","year":"2012","unstructured":"Bacuta, C., Monk, P.: Multilevel discretization of symmetric saddle point systems without the discrete LBB condition. Appl. Numer. Math. 62(6), 667\u2013681 (2012)","journal-title":"Appl. Numer. Math."},{"key":"697_CR4","doi-asserted-by":"crossref","unstructured":"Bacuta, C., Shu, L.: Multilevel gradient Uzawa algorithms for symmetric saddle point problems. J. Sci. Comput. 57, 105\u2013123 (2013)","DOI":"10.1007\/s10915-013-9697-9"},{"key":"697_CR5","unstructured":"Bacuta, C.: Cascadic multilevel algorithms for saddle point systems. Math. Numer. Anal. (2013). arXiv:1305.2449v1"},{"issue":"3","key":"697_CR6","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1137\/S0895479801395458","volume":"24","author":"ZZ Bai","year":"2003","unstructured":"Bai, Z.Z., Golub, G.H., Ng, M.K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear sytems. SIAM J. Matrix Anal. Appl. 24(3), 603\u2013626 (2003)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"697_CR7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00211-004-0521-1","volume":"98","author":"ZZ Bai","year":"2004","unstructured":"Bai, Z.Z., Golub, G.H., Pan, J.-Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1\u201332 (2004)","journal-title":"Numer. Math."},{"key":"697_CR8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00211-005-0643-0","volume":"102","author":"ZZ Bai","year":"2005","unstructured":"Bai, Z.Z., Parlett, B.N., Wang, Z.-Q.: On generalized succesive overrelaxation methods for augmented linear systems. Numer. Math. 102, 1\u201338 (2005)","journal-title":"Numer. Math."},{"issue":"2","key":"697_CR9","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1137\/050623644","volume":"28","author":"ZZ Bai","year":"2006","unstructured":"Bai, Z.Z., Golub, G.H., Li, C.-K.: Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices. SIAM J. Sci. Comput. 28(2), 583\u2013603 (2006)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"697_CR10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/imanum\/drl017","volume":"27","author":"ZZ Bai","year":"2007","unstructured":"Bai, Z.Z., Golub, G.H.: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal. 27(1), 1\u201323 (2007)","journal-title":"IMA J. Numer. Anal."},{"issue":"257","key":"697_CR11","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1090\/S0025-5718-06-01892-8","volume":"76","author":"ZZ Bai","year":"2007","unstructured":"Bai, Z.Z., Golub, G.H., Li, C.-K.: Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices. Math. Comput. 76(257), 287\u2013298 (2007)","journal-title":"Math. Comput."},{"key":"697_CR12","doi-asserted-by":"crossref","unstructured":"Bai, Z.-Z., Wang, Z.-Q.: On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl. 428(11\u201312), 2900\u20132932 (2008)","DOI":"10.1016\/j.laa.2008.01.018"},{"key":"697_CR13","doi-asserted-by":"crossref","unstructured":"Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1\u2013137 (2005)","DOI":"10.1017\/S0962492904000212"},{"issue":"3","key":"697_CR14","doi-asserted-by":"crossref","first-page":"1072","DOI":"10.1137\/S0036142994273343","volume":"34","author":"JH Bramble","year":"1997","unstructured":"Bramble, J.H., Pasciak, J.E., Vassilev, A.T.: Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34(3), 1072\u20131092 (1997)","journal-title":"SIAM J. Numer. Anal."},{"key":"697_CR15","doi-asserted-by":"crossref","unstructured":"Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial, 2nd edn. SIAM (2000)","DOI":"10.1137\/1.9780898719505"},{"issue":"6","key":"697_CR16","doi-asserted-by":"crossref","first-page":"1645","DOI":"10.1137\/0731085","volume":"31","author":"HC Elman","year":"1994","unstructured":"Elman, H.C., Golub, G.H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31(6), 1645\u20131661 (1994)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"697_CR17","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1137\/0723004","volume":"23","author":"HC Elman","year":"1986","unstructured":"Elman, H.C., Schultz, M.H.: Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations. SIAM J. Numer. Anal. 23(1), 44\u201357 (1986)","journal-title":"SIAM J. Numer. Anal."},{"key":"697_CR18","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1007\/s002110100300","volume":"90","author":"HC Elman","year":"2002","unstructured":"Elman, H.C., Silvester, D.J., Wathen, A.J.: Performance and analysis of saddle point preconditioners for the discrete steady-state Navier\u2013Stokes equations. Numer. Math. 90, 665\u2013688 (2002)","journal-title":"Numer. Math."},{"issue":"3","key":"697_CR19","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1016\/0378-4754(80)90053-1","volume":"22","author":"DJ Evans","year":"1980","unstructured":"Evans, D.J., Missirlis, N.M.: The preconditioned simultaneous displacement method (PSD method) for elliptic difference equations. Math. Comput. Simul. 22(3), 256\u2013263 (1980)","journal-title":"Math. Comput. Simul."},{"issue":"3","key":"697_CR20","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1007\/BF02510258","volume":"38","author":"B Fischer","year":"1998","unstructured":"Fischer, B., Ramage, R., Silvester, D.J., Wathen, A.J.: Minimum residual methods for augmented systems. BIT 38(3), 527\u2013543 (1998)","journal-title":"BIT"},{"issue":"1","key":"697_CR21","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1023\/A:1021965717530","volume":"41","author":"GH Golub","year":"2001","unstructured":"Golub, G.H., Wu, X., Yuan, J.-Y.: SOR-like methods for augmented systems. BIT 41(1), 71\u201385 (2001)","journal-title":"BIT"},{"issue":"8","key":"697_CR22","doi-asserted-by":"crossref","first-page":"1447","DOI":"10.1016\/j.cma.2006.03.020","volume":"196","author":"FJ Gaspar","year":"2007","unstructured":"Gaspar, F.J., Lisbona, F.J., Oosterlee, C.W., Vabishchevich, P.: An efficient multigrid solver for a reformulated version of the poroelasticity system. Comput. Methods Appl. Mech. Eng. 196(8), 1447\u20131457 (2007)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"3","key":"697_CR23","doi-asserted-by":"crossref","first-page":"1051","DOI":"10.1093\/imanum\/drr034","volume":"32","author":"A Gordon","year":"2012","unstructured":"Gordon, A., Powell, C.: On solving stochastic collocation systems with algebraic multigrid. IMA J. Numer. Anal. 32(3), 1051\u20131070 (2012)","journal-title":"IMA J. Numer. Anal."},{"key":"697_CR24","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719703","volume-title":"Numerical Simulation in Fluid Dynamics, A Practical Introduction","author":"M Griebel","year":"1998","unstructured":"Griebel, M., Dornseifer, T., Neunhoffer, T.: Numerical Simulation in Fluid Dynamics, A Practical Introduction. SIAM, Philadelphia (1998)"},{"issue":"2","key":"697_CR25","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1137\/S1064827502407810","volume":"25","author":"M Griebel","year":"2003","unstructured":"Griebel, M., Oeltz, D., Schweitzer, M.A.: An algebraic multigrid method for linear elasticity. SIAM J. Sci. Comput. 25(2), 385\u2013407 (2003)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2\u20133","key":"697_CR26","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1002\/nla.482","volume":"13","author":"M Griebel","year":"2006","unstructured":"Griebel, M., Metsch, B., Oeltz, D., Schweitzer, M.A.: Coarse grid classification: a parallel coarsening scheme for algebraic multigrid methods. Numer. Linear Algebra Appl. 13(2\u20133), 193\u2013214 (2006)","journal-title":"Numer. Linear Algebra Appl."},{"key":"697_CR27","unstructured":"Griebel, M., Metsch, B., Schweitzer, M.A.: Coarse grid classification\u2014Part II: automatic coarse grid agglomeration for parallel AMG. Preprint 271, Sonderforschungsbereich 611, Universit $$\\ddot{a}$$ a \u00a8 t Bonn (2006)"},{"key":"697_CR28","unstructured":"Griebel, M., Metsch, B., Schweitzer, M.A.: Coarse grid classification: AMG on parallel computers. In: M $$\\ddot{u}$$ u \u00a8 unster, G., Wolf, D., Kremer, M. (eds.) NIC Symposium 2008. NIC Series, vol. 39, pp. 299\u2013306 (2008)"},{"key":"697_CR29","doi-asserted-by":"crossref","unstructured":"Hadjidimos, A.: Accelerated overrelaxation methods. Math. Comput. 32(141), 149\u2013157 (1978)","DOI":"10.1090\/S0025-5718-1978-0483340-6"},{"key":"697_CR30","doi-asserted-by":"crossref","unstructured":"Hamilton, S., Benzi, M., Haber, E.: New multigrid smoothers for the Oseen problem. Numer. Linear Algebra Appl. 17(2\u20133), 557\u2013576 (2010)","DOI":"10.1002\/nla.707"},{"issue":"6","key":"697_CR31","doi-asserted-by":"crossref","first-page":"409","DOI":"10.6028\/jres.049.044","volume":"49","author":"MR Hestenes","year":"1952","unstructured":"Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49(6), 409\u2013436 (1952)","journal-title":"J. Res. Natl. Bur. Stand."},{"issue":"2","key":"697_CR32","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1002\/nla.1925","volume":"21","author":"C Lu","year":"2014","unstructured":"Lu, C., Jiao, X., Missirlis, N.M.: A hybrid geometric + algebraic multigrid method with semi-iterative smoothers. Numer. Linear Algebra Appl. 21(2), 221\u2013238 (2014)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"4","key":"697_CR33","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1093\/imanum\/23.4.581","volume":"23","author":"CJ Li","year":"2003","unstructured":"Li, C.J., Li, Z., Evans, D.J., Zhang, T.: A note on an SOR-like method for augmented systems. IMA J. Numer. Anal. 23(4), 581\u2013592 (2003)","journal-title":"IMA J. Numer. Anal."},{"issue":"11","key":"697_CR34","doi-asserted-by":"crossref","first-page":"2798","DOI":"10.1016\/j.laa.2009.12.016","volume":"432","author":"MA Louka","year":"2010","unstructured":"Louka, M.A., Missirlis, N.M., Tzaferis, F.I.: Is modified PSD equivalent to modified SOR for two-cyclic matrices? Linear Algebra Appl. 432(11), 2798\u20132815 (2010)","journal-title":"Linear Algebra Appl."},{"key":"697_CR35","unstructured":"Metsch, B.: Algebraic multigrid (AMG) for saddle point systems. Ph.D thesis, Mathematisch Naturwissenschaftlichen Fakult\u00e4t, Universit\u00e4t Bonn (2013)"},{"issue":"4","key":"697_CR36","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1137\/0718037","volume":"18","author":"NM Missirlis","year":"1981","unstructured":"Missirlis, N.M., Evans, D.J.: On the convergence of some generalised preconditioned iterative methods. SIAM J. Numer. Anal. 18(4), 591\u2013596 (1981)","journal-title":"SIAM J. Numer. Anal."},{"key":"697_CR37","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1007\/BF01391419","volume":"45","author":"NM Missirlis","year":"1984","unstructured":"Missirlis, N.M.: Convergence theory of extrapolated iterative methods for a certain class of non-symmetric linear systems. Numer. Math. 45, 447\u2013458 (1984)","journal-title":"Numer. Math."},{"issue":"3","key":"697_CR38","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/0378-4754(84)90062-4","volume":"26","author":"NM Missirlis","year":"1984","unstructured":"Missirlis, N.M., Evans, D.J.: The modified preconditioned simultaneous displacement (MPSD) method. Math. Comput. Simul. 26(3), 257\u2013262 (1984)","journal-title":"Math. Comput. Simul."},{"issue":"6","key":"697_CR39","doi-asserted-by":"crossref","first-page":"1969","DOI":"10.1137\/S1064827599355153","volume":"21","author":"MF Murphy","year":"2000","unstructured":"Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 21(6), 1969\u20131972 (2000)","journal-title":"SIAM J. Sci. Comput."},{"issue":"12","key":"697_CR40","doi-asserted-by":"crossref","first-page":"1933","DOI":"10.1016\/j.apnum.2007.11.014","volume":"58","author":"CW Oosterlee","year":"2008","unstructured":"Oosterlee, C.W., Gaspar, F.J.: Multigrid relaxation methods for systems of saddle point type. Appl. Numer. Math. 58(12), 1933\u20131950 (2008)","journal-title":"Appl. Numer. Math."},{"issue":"3","key":"697_CR41","doi-asserted-by":"crossref","first-page":"856","DOI":"10.1137\/0907058","volume":"7","author":"Y Saad","year":"1986","unstructured":"Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856\u2013869 (1986)","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"697_CR42","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898718003","volume-title":"Iterative Methods for Sparse Linear Systems","author":"Y Saad","year":"2003","unstructured":"Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)","edition":"2"},{"key":"697_CR43","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s00791-008-0094-0","volume":"11","author":"V Schulz","year":"2008","unstructured":"Schulz, V., Wittum, G.: Transforming smoothers for PDE constraint optimization problems. Comput. Vis. Sci 11, 207\u2013219 (2008)","journal-title":"Comput. Vis. Sci"},{"key":"697_CR44","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1007\/s00211-008-0187-1","volume":"111","author":"R Simon","year":"2009","unstructured":"Simon, R., Zulehner, W.: On Schwarz-type smoothers for saddle point problems with applications to PDE-constrained optimization problems. Numer. Math. 111, 445\u2013468 (2009)","journal-title":"Numer. Math."},{"key":"697_CR45","doi-asserted-by":"crossref","unstructured":"Takacs, S., Zulehner, W.: Multigrid methods for elliptic optimal control problems with Neumann boundary conditions. In: Proceedings of ENUMATH 2009 the 8th European Conference on Numerical Mathematics and Advanced Applications, Uppsalla, pp. 855\u2013863 (2009)","DOI":"10.1007\/978-3-642-11795-4_92"},{"key":"697_CR46","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/s00791-011-0168-2","volume":"14","author":"S Takacs","year":"2011","unstructured":"Takacs, S., Zulehner, W.: Convergence analysis of multigrid methods with collective point smoothers for optimal control prolems. Comput. Vis. Sci. 14, 131\u2013141 (2011)","journal-title":"Comput. Vis. Sci."},{"key":"697_CR47","unstructured":"Trottenberg, U., Oosterlee, C.W., Schuller, A.: Multigrid. Academic Press, New York (2001)"},{"key":"697_CR48","doi-asserted-by":"crossref","unstructured":"Van der Vorst, H.A.: Iterative Krylov Methods for Large Linear Systems, vol. 13. Cambridge University Press, London (2003)","DOI":"10.1017\/CBO9780511615115"},{"key":"697_CR49","volume-title":"Matrix Iterative Analysis","author":"RS Varga","year":"1962","unstructured":"Varga, R.S.: Matrix Iterative Analysis. Prentice-Hall, Inc., Englewood Cliffs (1962)"},{"issue":"7","key":"697_CR50","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1002\/fld.1080","volume":"50","author":"J Volker","year":"2006","unstructured":"Volker, J.: On the efficiency of linearization schemes and coupled multigrid methods in a simulation of a 3D flow around a cylinder. Int. J. Numer. Methods Fluids 50(7), 845\u2013862 (2006)","journal-title":"Int. J. Numer. Methods Fluids"},{"issue":"1\u20133","key":"697_CR51","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/S0024-3795(01)00275-0","volume":"330","author":"CL Wang","year":"2001","unstructured":"Wang, C.L., Bai, Z.Z.: Sufficient conditions for the convergent splittings of non-Hermitian positive definite matrices. Linear Algebra Appl. 330(1\u20133), 215\u2013218 (2001)","journal-title":"Linear Algebra Appl."},{"key":"697_CR52","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1007\/s10915-013-9684-1","volume":"56","author":"M Wang","year":"2013","unstructured":"Wang, M., Chen, L.: Multigrid methods for the Stokes equations using distributive Gauss\u2013Seidel relaxation based on the least squares commutator. J. Sci. Comput. 56, 409\u2013431 (2013)","journal-title":"J. Sci. Comput."},{"key":"697_CR53","doi-asserted-by":"crossref","unstructured":"Wathen, A.J., Silvester, D.J.: Fast iterative solution of stabilized Stokes systems. Part I: using simple diagonal preconditioners. SIAM J. Numer. Anal. 30(3), 630\u2013649 (1993)","DOI":"10.1137\/0730031"},{"key":"697_CR54","volume-title":"Iterative Solution of Large Linear Systems","author":"DM Young","year":"1971","unstructured":"Young, D.M.: Iterative Solution of Large Linear Systems. Academic Press, New York (1971)"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-015-0697-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-015-0697-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-015-0697-6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,19]],"date-time":"2019-08-19T12:55:13Z","timestamp":1566219313000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-015-0697-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1,14]]},"references-count":54,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2015,11]]}},"alternative-id":["697"],"URL":"https:\/\/doi.org\/10.1007\/s00211-015-0697-6","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1,14]]}}}