{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T16:41:44Z","timestamp":1787503304432,"version":"build-2736575974"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2016,12,30]],"date-time":"2016-12-30T00:00:00Z","timestamp":1483056000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["STI 157\/4-1"],"award-info":[{"award-number":["STI 157\/4-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2017,7]]},"DOI":"10.1007\/s10915-016-0345-z","type":"journal-article","created":{"date-parts":[[2016,12,30]],"date-time":"2016-12-30T09:56:47Z","timestamp":1483091807000},"page":"81-96","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Nonuniformly Weighted Schwarz Smoothers for Spectral Element Multigrid"],"prefix":"10.1007","volume":"72","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6485-3825","authenticated-orcid":false,"given":"J\u00f6rg","family":"Stiller","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2016,12,30]]},"reference":[{"issue":"2","key":"345_CR1","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1002\/nla.1816","volume":"19","author":"P Bastian","year":"2012","unstructured":"Bastian, P., Blatt, M., Scheichl, R.: Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems. Numer. Linear Algebra Appl. 19(2), 367\u2013388 (2012)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"6\u20137","key":"345_CR2","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1002\/nla.295","volume":"9","author":"R Blaheta","year":"2002","unstructured":"Blaheta, R.: GPCG-generalized preconditioned CG method and its use with non-linear and non-symmetric displacement decomposition preconditioners. Numer. Linear Algebra Appl. 9(6\u20137), 527\u2013550 (2002)","journal-title":"Numer. Linear Algebra Appl."},{"key":"345_CR3","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/j.procs.2015.05.241","volume":"51","author":"H Bouwmeester","year":"2015","unstructured":"Bouwmeester, H., Dougherty, A., Knyazev, A.V.: Nonsymmetric preconditioning for conjugate gradient and steepest descent methods. Procedia Comput. Sci. 51, 276\u2013285 (2015). ISSN 1877-0509","journal-title":"Procedia Comput. Sci."},{"key":"345_CR4","volume-title":"Multigrid Methods. Pitman Research Notes in Mathematics Series","author":"J Bramble","year":"1995","unstructured":"Bramble, J.: Multigrid Methods. Pitman Research Notes in Mathematics Series, vol. 294. Longman Scientific & Technical, Harlow (1995)"},{"issue":"1","key":"345_CR5","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1002\/anac.200410019","volume":"2","author":"SC Brenner","year":"2005","unstructured":"Brenner, S.C., Zhao, J.: Convergence of multigrid algorithms for interior penalty methods. Appl. Num. Anal. Comp. Math. 2(1), 3\u201318 (2005)","journal-title":"Appl. Num. Anal. Comp. Math."},{"key":"345_CR6","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1006\/jcph.1995.1011","volume":"116","author":"W Couzy","year":"1995","unstructured":"Couzy, W., Deville, M.O.: A fast Schur complement method for the spectral element discretization of the incompressible Navier\u2013Stokes equations. J. Comput. Phys. 116, 135\u2013142 (1995)","journal-title":"J. Comput. Phys."},{"key":"345_CR7","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511546792","volume-title":"High-Order Methods for Incompressible Fluid Flow","author":"MO Deville","year":"2002","unstructured":"Deville, M.O., Fischer, P.F., Mund, E.H.: High-Order Methods for Incompressible Fluid Flow, vol. 1. Cambridge University Press, Cambridge (2002)"},{"key":"345_CR8","doi-asserted-by":"crossref","unstructured":"Fischer, P.F., Lottes, J.W.: Hybrid Schwarz-multigrid methods for the spectral element method: Extensions to Navier\u2013Stokes. In: Kornhuber, R., P\u00e9riaux, J., Widlund, O., Hoppe, R., Pironneau, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering Series, pp. 35\u201349. Springer, Berlin (2004)","DOI":"10.1007\/3-540-26825-1_3"},{"issue":"4","key":"345_CR9","doi-asserted-by":"crossref","first-page":"1305","DOI":"10.1137\/S1064827597323415","volume":"21","author":"GH Golub","year":"1999","unstructured":"Golub, G.H., Ye, Q.: Inexact preconditioned conjugate gradient method with inner-outer iteration. SIAM J. Sci. Comput. 21(4), 1305\u20131320 (1999). ISSN 10648275","journal-title":"SIAM J. Sci. Comput."},{"key":"345_CR10","doi-asserted-by":"crossref","first-page":"6011","DOI":"10.1016\/j.cma.2005.10.010","volume":"195","author":"JL Guermond","year":"2006","unstructured":"Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195, 6011\u20136045 (2006)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"345_CR11","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.jcp.2013.07.035","volume":"255","author":"L Haupt","year":"2013","unstructured":"Haupt, L., Stiller, J., Nagel, W.: A fast spectral element solver combining static condensation and multigrid techniques. J. Comput. Phys. 255, 384\u2013395 (2013)","journal-title":"J. Comput. Phys."},{"key":"345_CR12","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/0021-9991(88)90161-1","volume":"77","author":"W Heinrichs","year":"1988","unstructured":"Heinrichs, W.: Line relaxation for spectral multigrid methods. J. Comput. Phys. 77, 166\u2013182 (1988)","journal-title":"J. Comput. Phys."},{"issue":"4","key":"345_CR13","doi-asserted-by":"crossref","first-page":"2095","DOI":"10.1137\/090778523","volume":"33","author":"B Janssen","year":"2011","unstructured":"Janssen, B., Kanschat, G.: Adaptive multilevel methods with local smoothing for $$H^1$$ H 1 - and $$H^{\\rm curl}$$ H curl -conforming high order finite element methods. SIAM J. Sci. Comput. 33(4), 2095\u20132114 (2011)","journal-title":"SIAM J. Sci. Comput."},{"key":"345_CR14","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/0377-0427(88)90358-5","volume":"24","author":"EF Kaasschieter","year":"1988","unstructured":"Kaasschieter, E.F.: Preconditioned conjugate gradients for solving singular systems. J. Comput. Appl. Math. 24, 265\u2013275 (1988)","journal-title":"J. Comput. Appl. Math."},{"key":"345_CR15","doi-asserted-by":"crossref","first-page":"2437","DOI":"10.1016\/j.compstruc.2004.04.015","volume":"82","author":"G Kanschat","year":"2004","unstructured":"Kanschat, G.: Multilevel methods for discontinuous Galerkin FEM on locally refined meshes. Comput. Struct. 82, 2437\u20132445 (2004)","journal-title":"Comput. Struct."},{"key":"345_CR16","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780198528692.001.0001","volume-title":"Spectral\/hp Element Methods for Computational Fluid Dynamics","author":"GE Karniadakis","year":"2005","unstructured":"Karniadakis, G.E., Sherwin, S.J.: Spectral\/hp Element Methods for Computational Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2005)","edition":"2"},{"issue":"5","key":"345_CR17","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1002\/nla.544","volume":"15","author":"JK Kraus","year":"2008","unstructured":"Kraus, J.K., Tomar, S.K.: A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problems. Numer. Linear Algebra Appl. 15(5), 417\u2013438 (2008)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"2","key":"345_CR18","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s10915-007-9183-3","volume":"36","author":"S Loisel","year":"2008","unstructured":"Loisel, S., Nabben, R., Szyld, D.: On hybrid multigrid-Schwarz algorithms. J. Sci. Comput. 36(2), 165\u2013175 (2008)","journal-title":"J. Sci. Comput."},{"key":"345_CR19","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s10915-004-4787-3","volume":"24","author":"JW Lottes","year":"2005","unstructured":"Lottes, J.W., Fischer, P.F.: Hybrid multigrid\/Schwarz algorithms for the spectral element method. J. Sci. Comput. 24, 45\u201378 (2005)","journal-title":"J. Sci. Comput."},{"key":"345_CR20","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/BF01386067","volume":"6","author":"RE Lynch","year":"1964","unstructured":"Lynch, R.E., Rice, J.R., Thomas, D.H.: Direct solution of partial difference equations by tensor product methods. Numer. Math. 6, 185\u2013199 (1964)","journal-title":"Numer. Math."},{"key":"345_CR21","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/BF01065177","volume":"3","author":"Y Maday","year":"1988","unstructured":"Maday, Y., Munoz, R.: Spectral element multigrid. II. Theoretical justification. J. Sci. Comput. 3, 323\u2013353 (1988)","journal-title":"J. Sci. Comput."},{"key":"345_CR22","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1002\/nla.700","volume":"17","author":"WF Mitchell","year":"2010","unstructured":"Mitchell, W.F.: The hp-multigrid method applied to hp-adaptive refinement of triangular grids. Numer. Linear Algebr. 17, 211\u2013228 (2010)","journal-title":"Numer. Linear Algebr."},{"issue":"4","key":"345_CR23","doi-asserted-by":"crossref","first-page":"1444","DOI":"10.1137\/S1064827599362314","volume":"22","author":"Y Notay","year":"2000","unstructured":"Notay, Y.: Flexible conjugate gradients. SIAM J. Sci. Comput. 22(4), 1444\u20131460 (2000)","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"345_CR24","doi-asserted-by":"crossref","first-page":"2189","DOI":"10.1137\/060663465","volume":"29","author":"L Olson","year":"2007","unstructured":"Olson, L.: Algebraic multigrid preconditioning of high-order spectral elements for elliptic problems on a simplicial mesh. SIAM J. Sci. Comput. 29(5), 2189\u20132209 (2007)","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"345_CR25","first-page":"667","volume":"5","author":"R Pasquetti","year":"2009","unstructured":"Pasquetti, R., Rapetti, F.: p-Multigrid method for Fekete-Gauss spectral element approximations of elliptic problems. Commun. Comput. Phys. 5(5), 667\u2013682 (2009)","journal-title":"Commun. Comput. Phys."},{"key":"345_CR26","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/BF01061297","volume":"2","author":"E R\u00f8nquist","year":"1987","unstructured":"R\u00f8nquist, E., Patera, A.: Spectral element multigrid. I. Formulation and numerical results. J. Sci. Comput. 2, 389\u2013406 (1987)","journal-title":"J. Sci. Comput."},{"key":"345_CR27","volume-title":"Multigrid","author":"U Trottenberg","year":"2000","unstructured":"Trottenberg, U., Oosterlee, C.W., Sch\u00fcller, A.: Multigrid. Academic Press, Oxford (2000)"},{"key":"345_CR28","doi-asserted-by":"crossref","unstructured":"Varga, R.S.: Matrix Iterative Analysis. Springer Series in Computational Mathematics, vol. 27, 2nd ed. Springer, Berlin (2000)","DOI":"10.1007\/978-3-642-05156-2"},{"issue":"8","key":"345_CR29","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1002\/fld.3767","volume":"72","author":"ZJ Wang","year":"2013","unstructured":"Wang, Z.J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H., Kroll, N., May, G., Persson, P.-O., van Leer, B., Visbal, M.: High-order CFD methods: current status and perspective. Int. J. Numer. Methods Fluids 72(8), 811\u2013845 (2013). ISSN 1097-0363","journal-title":"Int. J. Numer. Methods Fluids"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-016-0345-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-016-0345-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-016-0345-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,28]],"date-time":"2020-09-28T07:12:04Z","timestamp":1601277124000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-016-0345-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,30]]},"references-count":29,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,7]]}},"alternative-id":["345"],"URL":"https:\/\/doi.org\/10.1007\/s10915-016-0345-z","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,12,30]]}}}