{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,7]],"date-time":"2025-12-07T13:02:45Z","timestamp":1765112565128},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/501100004281","name":"Narodowe Centrum Nauki","doi-asserted-by":"publisher","award":["UMO-2017\/25\/B\/ST6\/02771"],"award-info":[{"award-number":["UMO-2017\/25\/B\/ST6\/02771"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We present an overlapping domain decomposition iterative solver for linear systems resulting from the discretization of compressible viscous flows with the Discontinuous Petrov\u2013Galerkin (DPG) method in three dimensions.\nIt is a two-grid solver utilizing the solution on the auxiliary coarse grid and the standard block-Jacobi iteration on patches of elements defined by supports of the coarse mesh base shape functions.\nThe simple iteration defined in this way is used as a preconditioner for the conjugate gradient procedure.\nTheoretical analysis indicates that the condition number of the preconditioned system should be independent of the actual finite element mesh and the auxiliary coarse mesh, provided that they are quasiuniform.\nNumerical tests confirm this result.\nMoreover, they show that presence of strongly flattened or elongated elements does not slow the convergence.\nThe finite element mesh is subject to adaptivity, i.e. dividing the elements with large errors until a required accuracy is reached.\nThe auxiliary coarse mesh is adjusting to the nonuniform actual mesh.<\/jats:p>","DOI":"10.1515\/cmam-2022-0206","type":"journal-article","created":{"date-parts":[[2023,7,11]],"date-time":"2023-07-11T14:52:06Z","timestamp":1689087126000},"page":"141-172","source":"Crossref","is-referenced-by-count":1,"title":["An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier\u2013Stokes Equations in 3D"],"prefix":"10.1515","volume":"24","author":[{"given":"Waldemar","family":"Rachowicz","sequence":"first","affiliation":[{"name":"Faculty of Computer Science and Telecommunications , Cracow University of Technology , Cracow , Poland"}]},{"given":"Witold","family":"Cecot","sequence":"additional","affiliation":[{"name":"Faculty of Civil Engineering , Cracow University of Technology , Cracow , Poland"}]},{"given":"Adam","family":"Zdunek","sequence":"additional","affiliation":[{"name":"HB BerRit , Solhemsbackarna 73 , Spanga , Sweden"}]}],"member":"374","published-online":{"date-parts":[[2023,7,12]]},"reference":[{"key":"2024010712054074184_j_cmam-2022-0206_ref_001","doi-asserted-by":"crossref","unstructured":"P. R. Amestoy, I. S. Duff, J.-Y. L\u2019Excellent and J. Koster,\nA fully asynchronous multifrontal solver using distributed dynamic scheduling,\nSIAM J. Matrix Anal. Appl. 23 (2001), no. 1, 15\u201341.","DOI":"10.1137\/S0895479899358194"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_002","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka and W. C. Rheinboldt,\nError estimates for adaptive finite element computations,\nSIAM J. Numer. Anal. 15 (1978), no. 4, 736\u2013754.","DOI":"10.1137\/0715049"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_003","doi-asserted-by":"crossref","unstructured":"J. H. Bramble and J. Xu,\nSome estimates for a weighted \n                  \n                     \n                        \n                           L\n                           2\n                        \n                     \n                     \n                     L^{2}\n                  \n                projection,\nMath. Comp. 56 (1991), no. 194, 463\u2013476.","DOI":"10.1090\/S0025-5718-1991-1066830-3"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_004","unstructured":"J. E. Carter,\nNumerical solutions of the Navier\u2013Stokes equations for the supersonic laminar flow over a two-dimensional compression corner,\nNASA Technical Report TR R-385, NASA Langley Research Center, Hampton, 1972."},{"key":"2024010712054074184_j_cmam-2022-0206_ref_005","doi-asserted-by":"crossref","unstructured":"J. Chan, L. Demkowicz and R. Moser,\nA DPG method for steady viscous compressible flow,\nComput. & Fluids 98 (2014), 69\u201390.","DOI":"10.1016\/j.compfluid.2014.02.024"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_006","doi-asserted-by":"crossref","unstructured":"J. Chan, N. Heuer, T. Bui-Thanh and L. Demkowicz,\nA robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms,\nComput. Math. Appl. 67 (2014), no. 4, 771\u2013795.","DOI":"10.1016\/j.camwa.2013.06.010"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_007","doi-asserted-by":"crossref","unstructured":"L. Demkowicz and N. Heuer,\nRobust DPG method for convection-dominated diffusion problems,\nSIAM J. Numer. Anal. 51 (2013), no. 5, 2514\u20132537.","DOI":"10.1137\/120862065"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_008","doi-asserted-by":"crossref","unstructured":"L. Demkowicz, J. Kurtz, D. Pardo, M. Paszy\u0144ski, W. Rachowicz and A. Zdunek,\nComputing with \n                  \n                     \n                        \n                           \n                              h\n                              \u2062\n                              p\n                           \n                        \n                        \n                        hp\n                     \n                  \n               -Adaptive Finite Elements. Vol. 2. Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications,\nChapman & Hall\/CRC, Boca Raton, 2008.","DOI":"10.1201\/9781420011692"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_009","doi-asserted-by":"crossref","unstructured":"L. Demkowicz, P. Monk, L. Vardapetyan and W. Rachowicz,\nde Rham diagram for \n                  \n                     \n                        \n                           h\n                           \u2062\n                           p\n                        \n                     \n                     \n                     hp\n                  \n                finite element spaces,\nComput. Math. Appl. 39 (2000), no. 7\u20138, 29\u201338.","DOI":"10.1016\/S0898-1221(00)00062-6"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_010","doi-asserted-by":"crossref","unstructured":"L. Demkowicz, J. T. Oden, W. Rachowicz and O. Hardy,\nToward a universal \u210e-\ud835\udc5d adaptive finite element strategy. I. Constrained approximation and data structure,\nComput. Methods Appl. Mech. Engrg. 77 (1989), no. 1\u20132, 79\u2013112.","DOI":"10.1016\/0045-7825(89)90129-1"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_011","unstructured":"M. Dryja and O. B. Widlund,\nAdditive Schwarz methods for elliptic finite element problems in three dimensions,\nFifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Norfolk 1991),\nSociety for Industrial and Applied Mathematics, Philadelphia (1992), 3\u201318."},{"key":"2024010712054074184_j_cmam-2022-0206_ref_012","unstructured":"G. H. Golub and C. F. Van Loan,\nMatrix Computations,\nJohns Hopkins University, Baltimore, 1989."},{"key":"2024010712054074184_j_cmam-2022-0206_ref_013","doi-asserted-by":"crossref","unstructured":"S. Petrides and L. Demkowicz,\nAn adaptive multigrid solver for DPG methods with applications in linear acoustics and electromagnetics,\nComput. Math. Appl. 87 (2021), 12\u201326.","DOI":"10.1016\/j.camwa.2021.01.017"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_014","doi-asserted-by":"crossref","unstructured":"W. Rachowicz,\nAn overlapping domain decomposition preconditioner for an anisotropic \u210e-adaptive finite element method,\nComput. Methods Appl. Mech. Engrg. 127 (1995), no. 1\u20134, 269\u2013292.","DOI":"10.1016\/0045-7825(95)00857-7"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_015","doi-asserted-by":"crossref","unstructured":"W. Rachowicz, A. Zdunek and W. Cecot,\nA discontinuous Petrov\u2013Galerkin method for compressible Navier\u2013Stokes equations in three dimensions,\nComput. Math. Appl. 102 (2021), 113\u2013136.","DOI":"10.1016\/j.camwa.2021.10.010"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_016","doi-asserted-by":"crossref","unstructured":"N. V. Roberts and J. Chan,\nA geometric multigrid preconditioning strategy for DPG system matrices,\nComput. Math. Appl. 74 (2017), no. 8, 2018\u20132043.","DOI":"10.1016\/j.camwa.2017.06.055"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_017","doi-asserted-by":"crossref","unstructured":"H. A. Schwarz,\nUeber einige Abbildungsaufgaben,\nJ. Reine Angew. Math. 70 (1869), 105\u2013120.","DOI":"10.1515\/crll.1869.70.105"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_018","unstructured":"O. B. Widlund,\nSome Schwarz methods for symmetric and nonsymmetric elliptic problems,\nFifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Norfolk 1991),\nSociety for Industrial and Applied Mathematics, Philadelphia (1992), 19\u201336."},{"key":"2024010712054074184_j_cmam-2022-0206_ref_019","doi-asserted-by":"crossref","unstructured":"J. Xu,\nIterative methods by space decomposition and subspace correction,\nSIAM Rev. 34 (1992), no. 4, 581\u2013613.","DOI":"10.1137\/1034116"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_020","unstructured":"D. M. Young,\nIterative Solution of Large Linear Systems,\nAcademic Press, New York, 1971."},{"key":"2024010712054074184_j_cmam-2022-0206_ref_021","doi-asserted-by":"crossref","unstructured":"A. Zdunek,\nTests with FALKSOL: A massively parallel multi-level domain decomposing direct solver,\nComput. Math. Appl. 97 (2021), 207\u2013222.","DOI":"10.1016\/j.camwa.2021.06.001"},{"key":"2024010712054074184_j_cmam-2022-0206_ref_022","doi-asserted-by":"crossref","unstructured":"O. C. Zienkiewicz, R. L. Taylor and P. Nithiarasu,\nThe Finite Element Method for Fluid Dynamics, 7th ed.,\nButterworth-Heinemann, Oxford, 2013.","DOI":"10.1016\/B978-1-85617-635-4.00014-5"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0206\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0206\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,7]],"date-time":"2024-01-07T12:07:05Z","timestamp":1704629225000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0206\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,12]]},"references-count":22,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,5,23]]},"published-print":{"date-parts":[[2024,1,1]]}},"alternative-id":["10.1515\/cmam-2022-0206"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0206","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,12]]}}}