{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T09:19:52Z","timestamp":1768727992595,"version":"3.49.0"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T00:00:00Z","timestamp":1667865600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T00:00:00Z","timestamp":1667865600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004769","name":"Universit\u00e0 degli Studi di Pavia","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004769","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2022,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A novel theoretical convergence rate estimate for a Balancing Domain Decomposition by Constraints algorithm is proven for the solution of the cardiac bidomain model, describing the propagation of the electric impulse in the cardiac tissue. The non-linear system arises from a fully implicit time discretization and a monolithic solution approach. The preconditioned non-symmetric operator is constructed from the linearized system arising within the Newton\u2013Krylov approach for the solution of the non-linear problem; we theoretically analyze and prove a convergence rate bound for the Generalised Minimal Residual iterations\u2019 residual. The theory is confirmed by extensive parallel numerical tests, widening the class of robust and efficient solvers for implicit time discretizations of the bidomain model.<\/jats:p>","DOI":"10.1007\/s00211-022-01331-x","type":"journal-article","created":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T15:57:11Z","timestamp":1667923031000},"page":"841-879","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Newton\u2013Krylov-BDDC deluxe solvers for non-symmetric fully implicit time discretizations of the bidomain model"],"prefix":"10.1007","volume":"152","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3639-5844","authenticated-orcid":false,"given":"Ngoc Mai Monica","family":"Huynh","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,11,8]]},"reference":[{"key":"1331_CR1","unstructured":"Balay, S., et al.: PETSc web page. https:\/\/www.mcs.anl.gov\/petsc\/ (2019)"},{"issue":"3","key":"1331_CR2","doi-asserted-by":"publisher","first-page":"A1118","DOI":"10.1137\/130917399","volume":"36","author":"L Beir\u00e3o Da Veiga","year":"2014","unstructured":"Beir\u00e3o Da Veiga, L., Pavarino, L.F., Scacchi, S., Widlund, O., Zampini, S.: Isogeometric BDDC preconditioners with deluxe scaling. SIAM J. Sci. Comput. 36(3), A1118\u2013A1139 (2014)","journal-title":"SIAM J. Sci. Comput."},{"key":"1331_CR3","doi-asserted-by":"crossref","unstructured":"Carusi, A., Burrage, K., Rodr\u00edguez, B.: Bridging experiments, models and simulations: an integrative approach to validation in computational cardiac electrophysiology. Am. J. Physiol. Heart Circ. Physiol. (2012)","DOI":"10.1152\/ajpheart.01151.2011"},{"key":"1331_CR4","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1016\/j.cam.2017.03.017","volume":"321","author":"H Chen","year":"2017","unstructured":"Chen, H., Li, X., Wang, Y.: A splitting preconditioner for a block two-by-two linear system with applications to the bidomain equations. J. Comput. Appl. Math. 321, 487\u2013498 (2017)","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"1331_CR5","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1007\/s10092-019-0317-4","volume":"56","author":"H Chen","year":"2019","unstructured":"Chen, H., Li, X., Wang, Y.: A two-parameter modified splitting preconditioner for the bidomain equations. Calcolo 56(2), 21 (2019)","journal-title":"Calcolo"},{"key":"1331_CR6","doi-asserted-by":"crossref","unstructured":"Colli Franzone, P., Savar\u00e9, G.: Degenerate evolution systems modeling the cardiac electric field at micro-and macroscopic level. In: Evolution Equations, Semigroups and Functional Analysis, pp. 49\u201378. Springer (2002)","DOI":"10.1007\/978-3-0348-8221-7_4"},{"key":"1331_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-04801-7","volume-title":"Mathematical Cardiac Electrophysiology","author":"P Colli Franzone","year":"2014","unstructured":"Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Mathematical Cardiac Electrophysiology. Springer, Berlin (2014)"},{"key":"1331_CR8","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/j.mbs.2016.08.003","volume":"280","author":"P Colli Franzone","year":"2016","unstructured":"Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Joint influence of transmural heterogeneities and wall deformation on cardiac bioelectrical activity: a simulation study. Math. Biosci. 280, 71\u201386 (2016)","journal-title":"Math. Biosci."},{"issue":"9","key":"1331_CR9","doi-asserted-by":"publisher","DOI":"10.1063\/1.4999465","volume":"27","author":"P Colli Franzone","year":"2017","unstructured":"Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Effects of mechanical feedback on the stability of cardiac scroll waves: a bidomain electro-mechanical simulation study. Chaos 27(9), 093905 (2017)","journal-title":"Chaos"},{"key":"1331_CR10","doi-asserted-by":"publisher","first-page":"268","DOI":"10.3389\/fphys.2018.00268","volume":"9","author":"P Colli Franzone","year":"2018","unstructured":"Colli Franzone, P., Pavarino, L.F., Scacchi, S.: A numerical study of scalable cardiac electro-mechanical solvers on HPC architectures. Front. Physiol. 9, 268 (2018)","journal-title":"Front. Physiol."},{"key":"1331_CR11","doi-asserted-by":"publisher","DOI":"10.1016\/j.mbs.2019.108228","volume":"315","author":"P Colli Franzone","year":"2019","unstructured":"Colli Franzone, P., Gionti, V., Pavarino, L.F., Scacchi, S., Storti, C.: Role of infarct scar dimensions, border zone repolarization properties and anisotropy in the origin and maintenance of cardiac reentry. Math. Biosci. 315, 108228 (2019)","journal-title":"Math. Biosci."},{"issue":"18","key":"1331_CR12","doi-asserted-by":"publisher","first-page":"2769","DOI":"10.1161\/CIRCULATIONAHA.105.532614","volume":"112","author":"R Coronel","year":"2005","unstructured":"Coronel, R., Casini, S., Koopmann, T.T., Wilms-Schopman, F.J.G., Verkerk, A.O., Groot, J.R., Bhuiyan, Z., Bezzina, C.R., Veldkamp, M.W., Linnenbank, A.C.: Right ventricular fibrosis and conduction delay in a patient with clinical signs of Brugada syndrome: a combined electrophysiological, genetic, histopathologic, and computational study. Circulation 112(18), 2769\u20132777 (2005)","journal-title":"Circulation"},{"key":"1331_CR13","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1016\/j.media.2018.04.005","volume":"47","author":"C Corrado","year":"2018","unstructured":"Corrado, C., Williams, S., Karim, R., Plank, G., O\u2019Neill, M., Niederer, S.: A work flow to build and validate patient specific left atrium electrophysiology models from catheter measurements. Med. Image Anal. 47, 153\u2013163 (2018)","journal-title":"Med. Image Anal."},{"issue":"1133","key":"1331_CR14","first-page":"353","volume":"307","author":"D Di Francesco","year":"1985","unstructured":"Di Francesco, D., Noble, D.: A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Philos. Trans. R. Soc. Lond. 307(1133), 353\u2013398 (1985)","journal-title":"Philos. Trans. R. Soc. Lond."},{"issue":"2","key":"1331_CR15","doi-asserted-by":"publisher","first-page":"C163","DOI":"10.1137\/130912505","volume":"36","author":"T Dickopf","year":"2014","unstructured":"Dickopf, T., Krause, D., Krause, R., Potse, M.: Design and analysis of a lightweight parallel adaptive scheme for the solution of the monodomain equation. SIAM J. Sci. Comput. 36(2), C163\u2013C189 (2014)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1331_CR16","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1137\/S1064827502412887","volume":"25","author":"CR Dohrmann","year":"2003","unstructured":"Dohrmann, C.R.: A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput. 25(1), 246\u2013258 (2003)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"1331_CR17","doi-asserted-by":"publisher","first-page":"745","DOI":"10.1002\/cpa.21574","volume":"69","author":"CR Dohrmann","year":"2016","unstructured":"Dohrmann, C.R., Widlund, O.B.: A BDDC algorithm with deluxe scaling for three-dimensional H (curl) problems. Commun. Pure Appl. Math. 69(4), 745\u2013770 (2016)","journal-title":"Commun. Pure Appl. Math."},{"issue":"2","key":"1331_CR18","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1137\/0720023","volume":"20","author":"SC Eisenstat","year":"1983","unstructured":"Eisenstat, S.C., Elman, H.C., Schultz, M.H.: Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20(2), 345\u2013357 (1983)","journal-title":"SIAM J. Numer. Anal."},{"key":"1331_CR19","unstructured":"Falgout, R.D., Yang, U.M.: Hypre, high performance preconditioners: Users manual. Technical report, Lawrence Livermore National Laboratory (2006)"},{"issue":"7","key":"1331_CR20","doi-asserted-by":"publisher","first-page":"1523","DOI":"10.1002\/nme.76","volume":"50","author":"C Farhat","year":"2001","unstructured":"Farhat, C., Lesoinne, M., LeTallec, P., Pierson, K., Rixen, D.: FETI-DP: a dual\u2013primal unified FETI method-part I: a faster alternative to the two-level FETI method. Int. J. Numer. Methods Eng. 50(7), 1523\u20131544 (2001)","journal-title":"Int. J. Numer. Methods Eng."},{"issue":"6","key":"1331_CR21","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1016\/S0006-3495(61)86902-6","volume":"1","author":"R FitzHugh","year":"1961","unstructured":"FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1(6), 445\u2013466 (1961)","journal-title":"Biophys. J."},{"key":"1331_CR22","unstructured":"FitzHugh, R.: Mathematical models of excitation and propagation in nerve. Biol. Eng., pp. 1\u201385 (1969)"},{"issue":"2","key":"1331_CR23","doi-asserted-by":"publisher","first-page":"B224","DOI":"10.1137\/20M1353848","volume":"44","author":"NMM Huynh","year":"2022","unstructured":"Huynh, N.M.M., Pavarino, L.F., Scacchi, S.: Parallel Newton\u2013Krylov-BDDC and FETI-DP deluxe solvers for implicit time discretizations of the cardiac bidomain equations. SIAM J. Sci. Comput. 44(2), B224\u2013B249 (2022)","journal-title":"SIAM J. Sci. Comput."},{"issue":"6","key":"1331_CR24","doi-asserted-by":"publisher","first-page":"1461","DOI":"10.1007\/s00466-020-01912-3","volume":"66","author":"Y Jiang","year":"2020","unstructured":"Jiang, Y., Chen, R., Cai, X.-C.: A highly parallel implicit domain decomposition method for the simulation of the left ventricle on unstructured meshes. Comput. Mech. 66(6), 1461\u20131475 (2020)","journal-title":"Comput. Mech."},{"issue":"6","key":"1331_CR25","doi-asserted-by":"publisher","first-page":"C417","DOI":"10.1137\/16M1102495","volume":"39","author":"A Klawonn","year":"2017","unstructured":"Klawonn, A., Lanser, M., Rheinbach, O.: Nonlinear FETI-DP and BDDC methods: a unified framework and parallel results. SIAM J. Sci. Comput. 39(6), C417\u2013C451 (2017)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1331_CR26","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. ZAMM Z. Angew. Math. Mech. 90(1), 5\u201332 (2010)","journal-title":"ZAMM Z. Angew. Math. Mech."},{"issue":"2","key":"1331_CR27","doi-asserted-by":"publisher","first-page":"H571","DOI":"10.1152\/ajpheart.1995.269.2.H571","volume":"269","author":"IJ LeGrice","year":"1995","unstructured":"LeGrice, I.J., Smaill, B.H., Chai, L.Z., Edgar, S.G., Gavin, J.B., Hunter, P.J.: Laminar structure of the heart: ventricular myocyte arrangement and connective tissue architecture in the dog. Am. J. Physiol. Heart Circ. Physiol. 269(2), H571\u2013H582 (1995)","journal-title":"Am. J. Physiol. Heart Circ. Physiol."},{"issue":"2","key":"1331_CR28","doi-asserted-by":"publisher","first-page":"A1171","DOI":"10.1137\/17M1128502","volume":"40","author":"L Liu","year":"2018","unstructured":"Liu, L., Keyes, D.E., Krause, R.: A note on adaptive nonlinear preconditioning techniques. SIAM J. Sci. Comput. 40(2), A1171\u2013A1186 (2018)","journal-title":"SIAM J. Sci. Comput."},{"issue":"6","key":"1331_CR29","doi-asserted-by":"publisher","first-page":"1501","DOI":"10.1161\/01.RES.68.6.1501","volume":"68","author":"C Luo","year":"1991","unstructured":"Luo, C., Rudy, Y.: A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ. Res. 68(6), 1501\u20131526 (1991)","journal-title":"Circ. Res."},{"issue":"7","key":"1331_CR30","doi-asserted-by":"publisher","first-page":"639","DOI":"10.1002\/nla.341","volume":"10","author":"J Mandel","year":"2003","unstructured":"Mandel, J., Dohrmann, C.R.: Convergence of a balancing domain decomposition by constraints and energy minimization. Numer. Linear Algebra Appl. 10(7), 639\u2013659 (2003)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"2","key":"1331_CR31","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1016\/j.apnum.2004.09.022","volume":"54","author":"J Mandel","year":"2005","unstructured":"Mandel, J., Dorhmann, C.R., Tezaur, R.: An algebraic theory for primal and dual substructuring methods by constraints. Appl. Numer. Math. 54(2), 167\u2013193 (2005)","journal-title":"Appl. Numer. Math."},{"key":"1331_CR32","doi-asserted-by":"publisher","first-page":"356","DOI":"10.3389\/fphys.2018.00356","volume":"9","author":"C Mendonca Costa","year":"2018","unstructured":"Mendonca Costa, C., Plank, G., Rinaldi, C.A., Niederer, S., Bishop, M.J.: Modeling the electrophysiological properties of the infarct border zone. Front. Physiol. 9, 356 (2018)","journal-title":"Front. Physiol."},{"issue":"7","key":"1331_CR33","doi-asserted-by":"publisher","first-page":"1065","DOI":"10.1142\/S0218202509003723","volume":"19","author":"M Munteanu","year":"2009","unstructured":"Munteanu, M., Pavarino, L.F.: Decoupled Schwarz algorithms for implicit discretizations of nonlinear monodomain and bidomain systems. Math. Models Methods Appl. Sci. 19(7), 1065\u20131097 (2009)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"5","key":"1331_CR34","doi-asserted-by":"publisher","first-page":"3861","DOI":"10.1137\/08074355X","volume":"31","author":"M Munteanu","year":"2009","unstructured":"Munteanu, M., Pavarino, L.F., Scacchi, S.: A scalable Newton\u2013Krylov\u2013Schwarz method for the bidomain reaction\u2013diffusion system. SIAM J. Sci. Comput. 31(5), 3861\u20133883 (2009)","journal-title":"SIAM J. Sci. Comput."},{"key":"1331_CR35","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1002\/nla.381","volume":"11","author":"M Murillo","year":"2004","unstructured":"Murillo, M., Cai, X.-C.: A fully implicit parallel algorithm for simulating the non-linear electrical activity of the heart. Numer. Linear Algebra Appl. 11, 261\u2013277 (2004)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"2","key":"1331_CR36","doi-asserted-by":"publisher","first-page":"100","DOI":"10.1038\/s41569-018-0104-y","volume":"16","author":"SA Niederer","year":"2019","unstructured":"Niederer, S.A., Lumens, J., Trayanova, N.A.: Computational models in cardiology. Nature Rev. Cardiol. 16(2), 100\u2013111 (2019)","journal-title":"Nature Rev. Cardiol."},{"key":"1331_CR37","doi-asserted-by":"publisher","first-page":"562","DOI":"10.1016\/j.cma.2015.07.009","volume":"295","author":"LF Pavarino","year":"2015","unstructured":"Pavarino, L.F., Scacchi, S., Zampini, S.: Newton\u2013Krylov-BDDC solvers for nonlinear cardiac mechanics. Comput. Methods Appl. Mech. Eng. 295, 562\u2013580 (2015)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"4","key":"1331_CR38","doi-asserted-by":"publisher","first-page":"1333","DOI":"10.1137\/040615249","volume":"37","author":"M Pennacchio","year":"2005","unstructured":"Pennacchio, M., Savar\u00e9, G., Colli Franzone, P.: Multiscale modeling for the bioelectric activity of the heart. SIAM J. Math. Anal. 37(4), 1333\u20131370 (2005)","journal-title":"SIAM J. Math. Anal."},{"issue":"4","key":"1331_CR39","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1109\/TBME.2006.889181","volume":"54","author":"G Plank","year":"2007","unstructured":"Plank, G., et al.: Algebraic multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 54(4), 585\u2013596 (2007)","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"1331_CR40","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1016\/j.cma.2016.05.031","volume":"314","author":"A Quarteroni","year":"2017","unstructured":"Quarteroni, A., Lassila, T., Rossi, S., Ruiz-Baier, R.: Integrated heart-coupling multiscale and multiphysics models for the simulation of the cardiac function. Comput. Methods Appl. Mech. Eng. 314, 345\u2013407 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"8","key":"1331_CR41","doi-asserted-by":"publisher","first-page":"743","DOI":"10.1109\/10.310090","volume":"41","author":"JM Rogers","year":"1994","unstructured":"Rogers, J.M., McCulloch, A.D.: A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Trans. Biomed. Eng. 41(8), 743\u2013757 (1994)","journal-title":"IEEE Trans. Biomed. Eng."},{"issue":"3","key":"1331_CR42","doi-asserted-by":"publisher","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":"1331_CR43","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1016\/j.cma.2010.09.016","volume":"200","author":"S Scacchi","year":"2011","unstructured":"Scacchi, S.: A multilevel hybrid Newton\u2013Krylov\u2013Schwarz method for the Bidomain model of electrocardiology. Comput. Methods Appl. Mech. Eng. 200, 717\u2013725 (2011)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"2","key":"1331_CR44","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1016\/j.mbs.2005.01.001","volume":"194","author":"J Sundnes","year":"2005","unstructured":"Sundnes, J., Lines, G.T., Tveito, A.: An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso. Math. Biosci. 194(2), 233\u2013248 (2005)","journal-title":"Math. Biosci."},{"issue":"4","key":"1331_CR45","doi-asserted-by":"publisher","first-page":"H1573","DOI":"10.1152\/ajpheart.00794.2003","volume":"286","author":"KHWJ Ten Tusscher","year":"2004","unstructured":"Ten Tusscher, K.H.W.J., Noble, D., Noble, P.-J., Panfilov, A.V.: A model for human ventricular tissue. Am. J. Physiol. Heart Circ. Physiol. 286(4), H1573\u2013H1589 (2004)","journal-title":"Am. J. Physiol. Heart Circ. Physiol."},{"key":"1331_CR46","volume-title":"Domain Decomposition Methods-Algorithms and Theory","author":"A Toselli","year":"2006","unstructured":"Toselli, A., Widlund, O.: Domain Decomposition Methods-Algorithms and Theory. Springer, Berlin (2006)"},{"issue":"1","key":"1331_CR47","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1161\/CIRCRESAHA.110.223610","volume":"108","author":"NA Trayanova","year":"2011","unstructured":"Trayanova, N.A.: Whole-heart modeling: applications to cardiac electrophysiology and electromechanics. Circ. Res. 108(1), 113\u2013128 (2011)","journal-title":"Circ. Res."},{"issue":"1","key":"1331_CR48","doi-asserted-by":"publisher","first-page":"25","DOI":"10.2140\/camcos.2008.3.25","volume":"3","author":"X Tu","year":"2008","unstructured":"Tu, X., Li, J.: A balancing domain decomposition method by constraints for advection\u2013diffusion problems. Commun. Appl. Math. Comput. Sci. 3(1), 25\u201360 (2008)","journal-title":"Commun. Appl. Math. Comput. Sci."},{"issue":"4","key":"1331_CR49","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1142\/S0218202513500632","volume":"24","author":"S Zampini","year":"2014","unstructured":"Zampini, S.: Dual\u2013primal methods for the cardiac Bidomain model. Math. Models Methods Appl. Sci. 24(4), 667\u2013696 (2014)","journal-title":"Math. Models Methods Appl. Sci."},{"key":"1331_CR50","doi-asserted-by":"crossref","unstructured":"Zampini, S.: Inexact BDDC methods for the cardiac bidomain model. In: Domain Decomposition Methods in Science and Engineering XXI, pp. 247\u2013255. Springer (2014)","DOI":"10.1007\/978-3-319-05789-7_21"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-022-01331-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-022-01331-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-022-01331-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,18]],"date-time":"2022-11-18T12:05:18Z","timestamp":1668773118000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-022-01331-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,8]]},"references-count":50,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,12]]}},"alternative-id":["1331"],"URL":"https:\/\/doi.org\/10.1007\/s00211-022-01331-x","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,8]]},"assertion":[{"value":"16 February 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 March 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 June 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}