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In addition to the standard no-net-flux constraints on each face, adaptive primal constraints obtained from the solutions of local generalized eigenvalue problems are included to control the condition number. Special deluxe scaling and local generalized eigenvalue problems are designed in order to make sure that these additional primal variables lie in a benign subspace in which the preconditioned operator is positive definite. The current multilevel theory for BDDC methods for porous media flow is complemented with an efficient algorithm for the computation of the so-called malign part of the solution, which is needed to make sure the rest of the solution can be obtained using the conjugate gradient iterates lying in the benign subspace. We also propose a new technique, based on the Sherman--Morrison formula, that lets us preserve the complexity of the subdomain local solvers. Condition number estimates are provided under certain standard assumptions. Extensive numerical experiments confirm the theoretical estimates; additional numerical results prove the effectiveness of the method with higher order elements and high-contrast problems from real-world applications.<\/jats:p>","DOI":"10.1137\/16m1080653","type":"journal-article","created":{"date-parts":[[2017,8,3]],"date-time":"2017-08-03T13:45:13Z","timestamp":1501767913000},"page":"A1389-A1415","source":"Crossref","is-referenced-by-count":37,"title":["Multilevel Balancing Domain Decomposition by Constraints Deluxe Algorithms with Adaptive Coarse Spaces for Flow in Porous Media"],"prefix":"10.1137","volume":"39","author":[{"given":"Stefano","family":"Zampini","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xuemin","family":"Tu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,8,3]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899358194"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(69)90200-5"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142903431924"},{"key":"atypb4","volume-title":"Argonne National Laboratory","author":"Balay S.","year":"2015"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/130917399"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/15M1054675"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-36519-5"},{"key":"atypb8","volume-title":"Technical report TR2015-979","author":"Calvo J. 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Goldfeld,\n                      Balancing Neumann-Neumann Preconditioners for the Mixed Formulation of Almost-Incompressible Linear Elasticity\n                      . Ph.D. thesis, New York University, Courant Institute of Mathematical Sciences, New York, 2003.","DOI":"10.1007\/s00211-002-0450-9"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-8191(01)00141-7"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/090766176"},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"G. Karypis,\n                      METIS and ParMETIS\n                      , in Encyclopedia of Parallel Computing, D. Padua, ed., Springer, New York, 2011, pp. 1117-1124.","DOI":"10.1007\/978-0-387-09766-4_500"},{"key":"atypb23","volume-title":"BDDC and FETI-DP Algorithms with Adaptive Coarse Spaces for Three-Dimensional Elliptic Problems with Oscillatory and High Contrast Coefficients, preprint, arXiv:1606.07560","author":"Kim H. 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