{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T20:53:50Z","timestamp":1761598430295,"version":"3.37.3"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,4,18]],"date-time":"2022-04-18T00:00:00Z","timestamp":1650240000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,4,18]],"date-time":"2022-04-18T00:00:00Z","timestamp":1650240000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001823","name":"Ministerstvo \u0160kolstv\u00ed, Ml\u00e1de\u017ee a T\u011blov\u00fdchovy","doi-asserted-by":"publisher","award":["LQ1602"],"award-info":[{"award-number":["LQ1602"]}],"id":[{"id":"10.13039\/501100001823","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001823","name":"Ministerstvo \u0160kolstv\u00ed, Ml\u00e1de\u017ee a T\u011blov\u00fdchovy","doi-asserted-by":"publisher","award":["e-INFRA CZ\u2013 LM2018140"],"award-info":[{"award-number":["e-INFRA CZ\u2013 LM2018140"]}],"id":[{"id":"10.13039\/501100001823","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001824","name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky","doi-asserted-by":"publisher","award":["19-11441S"],"award-info":[{"award-number":["19-11441S"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2022,10]]},"DOI":"10.1007\/s11075-022-01281-3","type":"journal-article","created":{"date-parts":[[2022,4,18]],"date-time":"2022-04-18T06:02:41Z","timestamp":1650261761000},"page":"773-801","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Highly scalable hybrid domain decomposition method for the solution of huge scalar variational inequalities"],"prefix":"10.1007","volume":"91","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3252-4795","authenticated-orcid":false,"given":"Zden\u011bk","family":"Dost\u00e1l","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Hor\u00e1k","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jakub","family":"Kru\u017e\u00edk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tom\u00e1\u0161","family":"Brzobohat\u00fd","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Old\u0159ich","family":"Vlach","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,4,18]]},"reference":[{"key":"1281_CR1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511624100","volume-title":"Iterative Solution Methods","author":"O Axelsson","year":"1994","unstructured":"Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1994)"},{"key":"1281_CR2","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1007\/s002110050446","volume":"83","author":"SC Brenner","year":"1999","unstructured":"Brenner, S. C.: The condition number of the Schur complement. Numer. Math. 83, 187\u2013203 (1999)","journal-title":"Numer. Math."},{"issue":"5","key":"1281_CR3","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1002\/nme.3187","volume":"88","author":"T Brzobohat\u00fd","year":"2011","unstructured":"Brzobohat\u00fd, T., Dost\u00e1l, Z., Kozubek, T., Kov\u00e1\u0159, P., Markopoulos, A.: Cholesky decomposition with fixing nodes to the stable computation of a generalized inverse of the stiffness matrix of a floating structure. Int. J. Numer. Methods Eng. 88(5), 493\u2013509 (2011)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"1281_CR4","unstructured":"Brzobohat\u00fd, T., Jaro\u0161ov\u00e1, M., Kozubek, T., Men\u0161\u00edk, M., Markopoulos, A.: The hybrid total FETI method. In: Proceedings of the Third International Conference on Parallel, Distributed, Grid, and Cloud Computing for Engineering Civil-Comp (2013)"},{"key":"1281_CR5","doi-asserted-by":"publisher","first-page":"545","DOI":"10.1137\/0728030","volume":"28","author":"AR Conn","year":"1991","unstructured":"Conn, A. R., Gould, N. I. M., Toint, P. L.: A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28, 545\u2013572 (1991)","journal-title":"SIAM J. Numer. Anal."},{"key":"1281_CR6","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1080\/00207168808803625","volume":"23","author":"Z Dost\u00e1l","year":"1988","unstructured":"Dost\u00e1l, Z.: Conjugate gradient method with preconditioning by projector. Int. J. Comput. Math. 23, 315\u2013324 (1988)","journal-title":"Int. J. Comput. Math."},{"issue":"2\u20134","key":"1281_CR7","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1023\/B:NUMA.0000005347.98806.b2","volume":"34","author":"Z Dost\u00e1l","year":"2003","unstructured":"Dost\u00e1l, Z.: A proportioning based algorithm with rate of convergence for bound constrained quadratic programming. Numer. Algorithms 34(2\u20134), 293\u2013302 (2003)","journal-title":"Numer. Algorithms"},{"key":"1281_CR8","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1007\/s00607-006-0184-0","volume":"78","author":"Z Dost\u00e1l","year":"2006","unstructured":"Dost\u00e1l, Z.: An optimal algorithm for bound and equality constrained quadratic programming problems with bounded spectrum. Computing 78, 311\u2013328 (2006)","journal-title":"Computing"},{"key":"1281_CR9","first-page":"25","volume":"31","author":"Z Dost\u00e1l","year":"2008","unstructured":"Dost\u00e1l, Z.: On the decrease of a quadratic function along the projected\u2013gradient path. ETNA 31, 25\u201359 (2008)","journal-title":"ETNA"},{"key":"1281_CR10","volume-title":"Optimal quadratic programming algorithms, with applications to variational inequalities, 1st edn","author":"Z Dost\u00e1l","year":"2009","unstructured":"Dost\u00e1l, Z.: Optimal quadratic programming algorithms, with applications to variational inequalities, 1st edn. Springer, New York (2009)"},{"issue":"2","key":"1281_CR11","doi-asserted-by":"publisher","first-page":"500","DOI":"10.1137\/050639454","volume":"45","author":"Z Dost\u00e1l","year":"2007","unstructured":"Dost\u00e1l, Z., Hor\u00e1k, D.: Theoretically supported scalable FETI for numerical solution of variational inequalities. SIAM J. Numer. Anal. 45(2), 500\u2013513 (2007)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"1281_CR12","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1023\/B:COAP.0000049888.80264.25","volume":"30","author":"Z Dost\u00e1l","year":"2005","unstructured":"Dost\u00e1l, Z., Sch\u00f6berl, J.: Minimizing quadratic functions subject to bound constraints with the rate of convergence and finite termination. Comput. Opt. Appl. 30(1), 23\u201344 (2005)","journal-title":"Comput. Opt. Appl."},{"issue":"1","key":"1281_CR13","doi-asserted-by":"publisher","first-page":"113565","DOI":"10.1016\/j.cam.2021.113565","volume":"394","author":"Z Dost\u00e1l","year":"2021","unstructured":"Dost\u00e1l, Z., Vlach, O.: An accelerated augmented Lagrangian algorithm with adaptive orthogonalization strategy for bound and equality constrained quadratic programming and its application to large-scale contact problems of elasticity. J. Comput. Appl. Math. 394(1), 113565 (2021)","journal-title":"J. Comput. Appl. Math."},{"issue":"1\u20132","key":"1281_CR14","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1016\/S0377-0427(99)00368-4","volume":"126","author":"Z Dost\u00e1l","year":"2000","unstructured":"Dost\u00e1l, Z., Gomes, F.A.M., Santos, S.A.: Duality based domain decomposition with natural coarse space for variational inequalities. J. Comput. Appl. Math. 126(1\u20132), 397\u2013415 (2000)","journal-title":"J. Comput. Appl. Math."},{"issue":"4","key":"1281_CR15","doi-asserted-by":"publisher","first-page":"1120","DOI":"10.1137\/S1052623499362573","volume":"13","author":"Z Dost\u00e1l","year":"2003","unstructured":"Dost\u00e1l, Z., Friedlander, A., Santos, SA: Augmented Lagrangians with adaptive precision control for quadratic programming with simple bounds and equality constraints. SIAM J. Optim. 13(4), 1120\u20131140 (2003)","journal-title":"SIAM J. Optim."},{"key":"1281_CR16","doi-asserted-by":"publisher","first-page":"1155","DOI":"10.1002\/cnm.881","volume":"22","author":"Z Dost\u00e1l","year":"2006","unstructured":"Dost\u00e1l, Z., Hor\u00e1k, D., Ku\u010dera, R.: Total FETI\u2014an easier implementable variant of the FETI method for numerical solution of elliptic PDE. Commun. Numer. Methods Eng. 22, 1155\u20131162 (2006)","journal-title":"Commun. Numer. Methods Eng."},{"issue":"6","key":"1281_CR17","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1002\/nla.1994","volume":"22","author":"Z Dost\u00e1l","year":"2015","unstructured":"Dost\u00e1l, Z., Kozubek, T., Vlach, O.: Reorthogonalization based stiffness preconditioning in FETI algorithms with applications to variational inequalities. Numer. Lin. Algebra Appl. 22(6), 987\u2013998 (2015)","journal-title":"Numer. Lin. Algebra Appl."},{"key":"1281_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4939-6834-3","volume-title":"Scalable Algorithms for Contact Problems AMM, vol. 36","author":"Z Dost\u00e1l","year":"2016","unstructured":"Dost\u00e1l, Z., Kozubek, T., Sadowsk\u00e1, M., Vondr\u00e1k, V.: Scalable Algorithms for Contact Problems AMM, vol. 36. Springer, New York (2016)"},{"key":"1281_CR19","first-page":"237","volume":"15","author":"Z Dost\u00e1l","year":"2017","unstructured":"Dost\u00e1l, Z., Hor\u00e1k, D., Sojka, R.: On the efficient reconstruction of displacements in FETI methods for contact problems. Adv. Electr. Electron. Eng. 15, 237\u2013241 (2017)","journal-title":"Adv. Electr. Electron. Eng."},{"key":"1281_CR20","doi-asserted-by":"publisher","unstructured":"Dost\u00e1l, Z., Hor\u00e1k, D., Brzobohat\u00fd, T., Vodstr\u010dil, P.: Bounds on the spectra of Schur complements of large H-TFETI clusters for 2D Laplacian and applications. Numer. Lin. Agebra Appl. https:\/\/doi.org\/10.1002\/nla.2344https:\/\/doi.org\/10.1002\/nla.2344","DOI":"10.1002\/nla.2344 10.1002\/nla.2344"},{"key":"1281_CR21","doi-asserted-by":"crossref","unstructured":"Dost\u00e1l, Z, Brzobohat\u00fd, T, Vlach, O: Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems. J. Numer. Math. (2021)","DOI":"10.1515\/jnma-2020-0048"},{"key":"1281_CR22","unstructured":"ESPRESO\u2014Highly Parallel Framework for Engineering Applications. http:\/\/numbox.it4i.cz"},{"key":"1281_CR23","doi-asserted-by":"publisher","first-page":"1205","DOI":"10.1002\/nme.1620320604","volume":"32","author":"C Farhat","year":"1991","unstructured":"Farhat, C., Roux, F.-X.: A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Methods Eng. 32, 1205\u20131227 (1991)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"1281_CR24","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1137\/0913020","volume":"13","author":"C Farhat","year":"1992","unstructured":"Farhat, C., Roux, F.-X.: An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems. SIAM J. Sci. Comput. 13, 379\u2013396 (1992)","journal-title":"SIAM J. Sci. Comput."},{"key":"1281_CR25","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1016\/0045-7825(94)90068-X","volume":"115","author":"C Farhat","year":"1994","unstructured":"Farhat, C., Mandel, J., Roux, F. -X.: Optimal convergence properties of the FETI domain decomposition method. Comput. Methods Appl. Mech. Eng. 115, 365\u2013385 (1994)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"7\u20138","key":"1281_CR26","doi-asserted-by":"publisher","first-page":"687","DOI":"10.1002\/1099-1506(200010\/12)7:7\/8<687::AID-NLA219>3.0.CO;2-S","volume":"7","author":"C Farhat","year":"2000","unstructured":"Farhat, C., Lesoinne, M., Pierson, K.: A scalable dual-primal domain decomposition method. Numer. Lin. Algebra Appl. 7(7\u20138), 687\u2013714 (2000)","journal-title":"Numer. Lin. Algebra Appl."},{"key":"1281_CR27","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1048-1","volume-title":"Solution of variational inequalities in mechanics","author":"I Hlav\u00e1\u010dek","year":"1988","unstructured":"Hlav\u00e1\u010dek, I., Haslinger, J., Ne\u010das, J., Lov\u00ed\u0161ek, J.: Solution of variational inequalities in mechanics. Springer, Berlin (1988). Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)"},{"issue":"10","key":"1281_CR28","doi-asserted-by":"publisher","first-page":"1894","DOI":"10.1016\/j.matcom.2010.10.031","volume":"82","author":"M Jaro\u0161ov\u00e1","year":"2012","unstructured":"Jaro\u0161ov\u00e1, M., Klawonn, A., Rheinbach, O.: Projector preconditioning and transformation of basis in FETI-DP algorithms for contact problem. Math. Comput. Simul. 82(10), 1894\u20131907 (2012)","journal-title":"Math. Comput. Simul."},{"issue":"5","key":"1281_CR29","doi-asserted-by":"publisher","first-page":"1886","DOI":"10.1137\/050624364","volume":"28","author":"A Klawonn","year":"2006","unstructured":"Klawonn, A., Rheinbach, O.: A parallel implementation of dual-primal FETI methodsfor three dimensional linear elasticity using a transformation of basis. SIAM J. Sci. Comput. 28(5), 1886\u20131906 (2006)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1281_CR30","doi-asserted-by":"publisher","first-page":"10841","DOI":"10.1002\/pamm.200810841","volume":"90","author":"A Klawonn","year":"2008","unstructured":"Klawonn, A., Rheinbach, O.: A hybrid approach to 3-level FETI. Proc. Appl. Math. Mech. 90(1), 10841\u201310843 (2008)","journal-title":"Proc. Appl. Math. Mech."},{"issue":"1","key":"1281_CR31","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1002\/zamm.200900329","volume":"90","author":"A Klawonn","year":"2010","unstructured":"Klawonn, A., Rheinbach, O.: Highly scalable parallel domain decomposition methods with an application to biomechanics. Z. Angew. Math. Mech. 90(1), 5\u201332 (2010)","journal-title":"Z. Angew. Math. Mech."},{"key":"1281_CR32","first-page":"1523","volume":"LIX","author":"A Klawonn","year":"2006","unstructured":"Klawonn, A., Widlund, O.: Dual\u2013primal FETI method for linear elasticity. Commun. Pure Appl. Anal. LIX, 1523\u20131572 (2006)","journal-title":"Commun. Pure Appl. Anal."},{"key":"1281_CR33","doi-asserted-by":"publisher","first-page":"C667","DOI":"10.1137\/140997907","volume":"6","author":"A Klawonn","year":"2015","unstructured":"Klawonn, A., Lanser, M., Rheinbach, O.: Toward extremally scalable nonlinear domain decomposition methods for elliptic partiall differential equations. SIAM J. Sci. Comput. (37) 6, C667\u2013C696 (2015)","journal-title":"SIAM J. Sci. Comput. (37)"},{"issue":"1028","key":"1281_CR34","first-page":"95","volume":"149","author":"J Kru\u017e\u00edk","year":"2020","unstructured":"Kru\u017e\u00edk, J., Hor\u00e1k, D., \u010cerm\u00e1k, M., Posp\u00edsil, L., Pecha, M.: Active set expansion strategies in MPRGP algorithm. Adv. Eng. Softw. 149(1028), 95 (2020)","journal-title":"Adv. Eng. Softw."},{"key":"1281_CR35","doi-asserted-by":"crossref","unstructured":"Lee, J.: Domain decomposition methods for auxiliary linear problems of an elliptic variational inequality. In: Bank, R., et al (eds.) Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering, vol. 91, pp 319\u2013326 (2013)","DOI":"10.1007\/978-3-642-35275-1_35"},{"issue":"3","key":"1281_CR36","doi-asserted-by":"publisher","first-page":"1350","DOI":"10.1137\/100783753","volume":"35","author":"J Lee","year":"2013","unstructured":"Lee, J.: Two domain decomposition methods for auxiliary linear problems for a multibody variational inequality. SIAM J. Sci. Comput. 35(3), 1350\u20131375 (2013)","journal-title":"SIAM J. Sci. Comput."},{"key":"1281_CR37","doi-asserted-by":"publisher","first-page":"250","DOI":"10.1002\/nme.1553","volume":"66","author":"J Li","year":"2006","unstructured":"Li, J., Widdlund, O. B.: FETI-DP, BDDC,and block Cholesky method. Int. J. Num. Methods Eng. 66, 250\u2013271 (2006)","journal-title":"Int. J. Num. Methods Eng."},{"key":"1281_CR38","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-23588-7","volume-title":"Finite and boundary element tearing and interconnecting solvers for multiscale problems","author":"C Pechstein","year":"2013","unstructured":"Pechstein, C.: Finite and boundary element tearing and interconnecting solvers for multiscale problems. Springer, Heidelberg (2013)"},{"key":"1281_CR39","unstructured":"PERMON\u2014Parallel, efficient, robust, modular, object-oriented numerical software toolbox. http:\/\/permon.vsb.cz\/"},{"key":"1281_CR40","doi-asserted-by":"publisher","DOI":"10.1007\/b137868","volume-title":"Domain decomposition methods\u2014algorithms and theory Springer Series on Computational Mathematics, vol. 34","author":"A Toselli","year":"2005","unstructured":"Toselli, A., Widlund, O. B.: Domain decomposition methods\u2014algorithms and theory Springer Series on Computational Mathematics, vol. 34. Springer, Berlin (2005)"},{"issue":"6","key":"1281_CR41","doi-asserted-by":"publisher","first-page":"699","DOI":"10.21136\/AM.2017.0193-17","volume":"62","author":"P Vodstr\u010dil","year":"2017","unstructured":"Vodstr\u010dil, P., Bouchala, J., Jaro\u0161ov\u00e1, M., Dost\u00e1l, Z.: On conditioning of Schur complements of h-TFETI clusters for 2D problems governed by Laplacian. Appl. Math. 62(6), 699\u2013718 (2017)","journal-title":"Appl. Math."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-022-01281-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-022-01281-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-022-01281-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,9]],"date-time":"2022-09-09T07:42:57Z","timestamp":1662709377000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-022-01281-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,18]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["1281"],"URL":"https:\/\/doi.org\/10.1007\/s11075-022-01281-3","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2022,4,18]]},"assertion":[{"value":"18 May 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 February 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 April 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}]}}