{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,2]],"date-time":"2023-06-02T22:12:56Z","timestamp":1685743976341},"reference-count":31,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/100008932","name":"Centre for Numerical Analysis and Intelligent Software","doi-asserted-by":"crossref","award":["EP\/G036136\/1"],"id":[{"id":"10.13039\/100008932","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,10,1]]},"abstract":"Abstract<\/jats:title>\n Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of nonstandard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the nonstandard interface conditions are naturally defined at the boundary between elements. In this paper, we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with nonstandard boundary conditions. The full stability and convergence analysis\nof the discretisation method is presented, and the results are corroborated by numerical experiments.\nIn addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.<\/jats:p>","DOI":"10.1515\/cmam-2018-0005","type":"journal-article","created":{"date-parts":[[2018,3,22]],"date-time":"2018-03-22T22:15:52Z","timestamp":1521756952000},"page":"703-722","source":"Crossref","is-referenced-by-count":6,"title":["Hybrid Discontinuous Galerkin Discretisation and Domain Decomposition Preconditioners for the Stokes Problem"],"prefix":"10.1515","volume":"19","author":[{"given":"Gabriel R.","family":"Barrenechea","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics , University of Strathclyde , 26 Richmond Street, G1 1XH Glasgow , United Kingdom"}]},{"ORCID":"http:\/\/orcid.org\/0000-0003-2723-6913","authenticated-orcid":false,"given":"Micha\u0142","family":"Bosy","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics , University of Strathclyde , 26 Richmond Street, G1 1XH Glasgow , United Kingdom"}]},{"ORCID":"http:\/\/orcid.org\/0000-0002-5885-1903","authenticated-orcid":false,"given":"Victorita","family":"Dolean","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics , University of Strathclyde , 26 Richmond Street, G1 1XH Glasgow , United Kingdom , and Facult\u00e9 des Sciences, University C\u00f4te d\u2019Azur, Lab. J-A Dieudonn\u00e9, Parc Valrose, 06108 Nice Cedex 02, France"}]},{"given":"Fr\u00e9d\u00e9ric","family":"Nataf","sequence":"additional","affiliation":[{"name":"CNRS, INRIA , Laboratoire Jacques-Louis Lions , Sorbonne Universit\u00e9 , Universit\u00e9 Paris-Diderot SPC , \u00e9quipe Alpines, 75005 Paris , France"}]},{"given":"Pierre-Henri","family":"Tournier","sequence":"additional","affiliation":[{"name":"CNRS, INRIA , Laboratoire Jacques-Louis Lions , Sorbonne Universit\u00e9 , Universit\u00e9 Paris-Diderot SPC , \u00e9quipe Alpines, 75005 Paris , France"}]}],"member":"374","published-online":{"date-parts":[[2018,3,22]]},"reference":[{"key":"2021021301340080945_j_cmam-2018-0005_ref_001_w2aab3b7b1b1b6b1ab2b1b1Aa","unstructured":"B. Ayuso de Dios, F. Brezzi, L. D. Marini, J. Xu and L. Zikatanov,\nA simple preconditioner for a discontinuous Galerkin method for the Stokes problem,\nJ. Sci. Comput. 58 (2014), no. 3, 517\u2013547."},{"key":"2021021301340080945_j_cmam-2018-0005_ref_002_w2aab3b7b1b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"D. Boffi, F. Brezzi and M. Fortin,\nMixed Finite Element Methods and Applications,\nSpringer Ser. Comput. Math. 44,\nSpringer, Heidelberg, 2013.","DOI":"10.1007\/978-3-642-36519-5"},{"key":"2021021301340080945_j_cmam-2018-0005_ref_003_w2aab3b7b1b1b6b1ab2b1b3Aa","unstructured":"X.-C. Cai and M. Sarkis,\nA restricted additive Schwarz preconditioner for general sparse linear systems,\nSIAM J. Sci. Comput. 21 (1999), no. 2, 792\u2013797."},{"key":"2021021301340080945_j_cmam-2018-0005_ref_004_w2aab3b7b1b1b6b1ab2b1b4Aa","unstructured":"T. Cluzeau, V. Dolean, F. Nataf and A. Quadrat,\nPreconditionning techniques for systems of partial differential equations based on algebraic methods,\nTechnical Report 7953, INRIA, 2012, http:\/\/hal.inria.fr\/hal-00694468."},{"key":"2021021301340080945_j_cmam-2018-0005_ref_005_w2aab3b7b1b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"T. Cluzeau, V. Dolean, F. Nataf and A. Quadrat,\nSymbolic techniques for domain decomposition methods,\nDomain Decomposition Methods in Science and Engineering XX,\nSpringer, Berlin (2013), 27\u201338.","DOI":"10.1007\/978-3-642-35275-1_3"},{"key":"2021021301340080945_j_cmam-2018-0005_ref_006_w2aab3b7b1b1b6b1ab2b1b6Aa","unstructured":"B. Cockburn, D. A. Di Pietro and A. Ern,\nBridging the hybrid high-order and hybridizable discontinuous Galerkin methods,\nESAIM Math. Model. Numer. Anal. 50 (2016), no. 3, 635\u2013650."},{"key":"2021021301340080945_j_cmam-2018-0005_ref_007_w2aab3b7b1b1b6b1ab2b1b7Aa","unstructured":"B. Cockburn and J. Gopalakrishnan,\nThe derivation of hybridizable discontinuous Galerkin methods for Stokes flow,\nSIAM J. Numer. Anal. 47 (2009), no. 2, 1092\u20131125."},{"key":"2021021301340080945_j_cmam-2018-0005_ref_008_w2aab3b7b1b1b6b1ab2b1b8Aa","unstructured":"B. Cockburn, J. Gopalakrishnan and R. Lazarov,\nUnified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems,\nSIAM J. Numer. Anal. 47 (2009), no. 2, 1319\u20131365."},{"key":"2021021301340080945_j_cmam-2018-0005_ref_009_w2aab3b7b1b1b6b1ab2b1b9Aa","unstructured":"B. Cockburn, J. Gopalakrishnan, N. C. Nguyen, J. Peraire and F. J. Sayas,\nAnalysis of HDG methods for Stokes flow,\nMath. Comp. 80 (2011), no. 274, 723\u2013760."},{"key":"2021021301340080945_j_cmam-2018-0005_ref_010_w2aab3b7b1b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"D. A. Di Pietro and A. Ern,\nMathematical Aspects of Discontinuous Galerkin Methods,\nMath. Appl. (Berlin) 69,\nSpringer, Heidelberg, 2012.","DOI":"10.1007\/978-3-642-22980-0"},{"key":"2021021301340080945_j_cmam-2018-0005_ref_011_w2aab3b7b1b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"D. A. Di Pietro and A. Ern,\nA hybrid high-order locking-free method for linear elasticity on general meshes,\nComput. Methods Appl. Mech. Engrg. 283 (2015), 1\u201321.","DOI":"10.1016\/j.cma.2014.09.009"},{"key":"2021021301340080945_j_cmam-2018-0005_ref_012_w2aab3b7b1b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"V. Dolean, P. Jolivet and F. Nataf,\nAn Introduction to Domain Decomposition Methods. Algorithms, Theory, and Parallel Implementation,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2015.","DOI":"10.1137\/1.9781611974065"},{"key":"2021021301340080945_j_cmam-2018-0005_ref_013_w2aab3b7b1b1b6b1ab2b1c13Aa","unstructured":"V. Dolean and F. Nataf,\nA new domain decomposition method for the compressible Euler equations,\nESAIM Math. Model. Numer. 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Methods Eng. 13 (2006), no. 4, 515\u2013572."},{"key":"2021021301340080945_j_cmam-2018-0005_ref_020_w2aab3b7b1b1b6b1ab2b1c20Aa","unstructured":"F. Hecht,\nNew development in FreeFem++,\nJ. Numer. Math. 20 (2012), no. 3\u20134, 251\u2013265."},{"key":"2021021301340080945_j_cmam-2018-0005_ref_021_w2aab3b7b1b1b6b1ab2b1c21Aa","unstructured":"C. Lehrenfeld,\nHybrid discontinuous Galerkin methods for solving incompressible flow problems,\nDissertation, Rheinisch-Westf\u00e4lischen Technischen Hochschule Aachen, 2010."},{"key":"2021021301340080945_j_cmam-2018-0005_ref_022_w2aab3b7b1b1b6b1ab2b1c22Aa","doi-asserted-by":"crossref","unstructured":"C. Lehrenfeld and J. Sch\u00f6berl,\nHigh order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows,\nComput. Methods Appl. Mech. Engrg. 307 (2016), 339\u2013361.","DOI":"10.1016\/j.cma.2016.04.025"},{"key":"2021021301340080945_j_cmam-2018-0005_ref_023_w2aab3b7b1b1b6b1ab2b1c23Aa","unstructured":"I. 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Thomas,\nOptimized multiplicative, additive, and restricted additive Schwarz preconditioning,\nSIAM J. Sci. Comput. 29 (2007), no. 6, 2402\u20132425."},{"key":"2021021301340080945_j_cmam-2018-0005_ref_031_w2aab3b7b1b1b6b1ab2b1c31Aa","doi-asserted-by":"crossref","unstructured":"A. Toselli and O. Widlund,\nDomain Decomposition Methods \u2013 Algorithms and Theory,\nSpringer Ser. Comput. 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