{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T09:12:45Z","timestamp":1768468365550,"version":"3.49.0"},"reference-count":45,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2021,9,28]],"date-time":"2021-09-28T00:00:00Z","timestamp":1632787200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/501100001659","name":"German Research Foundation","doi-asserted-by":"crossref","award":["KO5206\/1-1 and KR4661\/2-1"],"award-info":[{"award-number":["KO5206\/1-1 and KR4661\/2-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2021,12,31]]},"abstract":"<jats:p>\n            This work presents the efficient, matrix-free finite-element library\n            <jats:monospace>hyper.deal<\/jats:monospace>\n            for solving partial differential equations in two up to six dimensions with high-order discontinuous Galerkin methods. It builds upon the low-dimensional finite-element library\n            <jats:monospace>deal.II<\/jats:monospace>\n            to create complex low-dimensional meshes and to operate on them individually. These meshes are combined via a tensor product on the fly, and the library provides new special-purpose highly optimized matrix-free functions exploiting domain decomposition as well as shared memory via\n            <jats:monospace>MPI-3.0<\/jats:monospace>\n            features. Both node-level performance analyses and strong\/weak-scaling studies on up to 147,456 CPU cores confirm the efficiency of the implementation. Results obtained with the library\n            <jats:monospace>hyper.deal<\/jats:monospace>\n            are reported for high-dimensional advection problems and for the solution of the Vlasov\u2013Poisson equation in up to six-dimensional phase space.\n          <\/jats:p>","DOI":"10.1145\/3469720","type":"journal-article","created":{"date-parts":[[2021,9,28]],"date-time":"2021-09-28T20:47:05Z","timestamp":1632862025000},"page":"1-34","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":13,"title":["hyper.deal: An Efficient, Matrix-free Finite-element Library for High-dimensional Partial Differential Equations"],"prefix":"10.1145","volume":"47","author":[{"given":"Peter","family":"Munch","sequence":"first","affiliation":[{"name":"Institute for Computational Mechanics, Department of Mechanical Engineering, Technical University of Munich, Germany and Institute of Material Systems Modeling, Helmholtz-Zentrum Hereon, Geesthacht, Germany"}]},{"given":"Katharina","family":"Kormann","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Plasma Physics, Germany, Department of Mathematics, Technical University of Munich, Germany, and Department of Information Technology, Uppsala University, Uppsala, Sweden"}]},{"given":"Martin","family":"Kronbichler","sequence":"additional","affiliation":[{"name":"Institute for Computational Mechanics, Department of Mechanical Engineering, Technical University of Munich, Germany and Department of Information Technology, Uppsala University, Uppsala, Sweden"}]}],"member":"320","published-online":{"date-parts":[[2021,9,28]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9991(03)00194-3"},{"key":"e_1_2_1_2_1","first-page":"9","article-title":"The FEniCS Project Version 1.5","volume":"3","author":"Aln\u00e6s Martin S.","year":"2015","unstructured":"Martin S. Aln\u00e6s , Jan Blechta , Johan Hake , August Johansson , Benjamin Kehlet , Anders Logg , Chris Richardson , Johannes Ring , Marie E. Rognes , and Garth N. Wells . 2015 . The FEniCS Project Version 1.5 . Arch. Numer. Soft. 3 , 100 (2015), 9 \u2013 23 . DOI:http:\/\/dx.doi.org\/10.11588\/ans.2015.100.20553 10.11588\/ans.2015.100.20553 Martin S. Aln\u00e6s, Jan Blechta, Johan Hake, August Johansson, Benjamin Kehlet, Anders Logg, Chris Richardson, Johannes Ring, Marie E. Rognes, and Garth N. Wells. 2015. The FEniCS Project Version 1.5. Arch. Numer. Soft. 3, 100 (2015), 9\u201323. DOI:http:\/\/dx.doi.org\/10.11588\/ans.2015.100.20553","journal-title":"Arch. Numer. Soft."},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2020.06.009"},{"key":"e_1_2_1_4_1","volume-title":"The deal","author":"Arndt Daniel","year":"2020","unstructured":"Daniel Arndt , Wolfgang Bangerth , Bruno Blais , Thomas C. Clevenger , Marc Fehling , Alexander V. Grayver , Timo Heister , Luca Heltai , Martin Kronbichler , Matthias Maier , Peter Munch , Jean-Paul Pelteret , Reza Rastak , Ignacio Thomas , Bruno Turcksin , Zhuoran Wang , and David Wells . 2020. The deal . II Library , Version 9.2. J. Numer. Math. 28, 3 ( 2020 ), 131\u2013146. DOI:http:\/\/dx.doi.org\/10.1515\/jnma-2020-0043 10.1515\/jnma-2020-0043 Daniel Arndt, Wolfgang Bangerth, Bruno Blais, Thomas C. Clevenger, Marc Fehling, Alexander V. Grayver, Timo Heister, Luca Heltai, Martin Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, Reza Rastak, Ignacio Thomas, Bruno Turcksin, Zhuoran Wang, and David Wells. 2020. The deal.II Library, Version 9.2. J. Numer. Math. 28, 3 (2020), 131\u2013146. DOI:http:\/\/dx.doi.org\/10.1515\/jnma-2020-0043"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2020.02.022"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-016-9317-9"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/2049673.2049678"},{"key":"e_1_2_1_8_1","volume-title":"Software for Exascale Computing \u2013 SPPEXA 2013\u20132015, Hans-Joachim Bungartz, Peter Neumann, and Wolfgang E","author":"Bastian Peter","unstructured":"Peter Bastian , Christian Engwer , Jorrit Fahlke , Markus Geveler , Dominik G\u00f6ddeke , Oleg Iliev , Olaf Ippisch , Ren\u00e9 Milk , Jan Mohring , Steffen M\u00fcthing , Mario Ohlberger , Dirk Ribbrock , and Stefan Turek . 2016. Hardware-based efficiency advances in the EXA-DUNE Project . In Software for Exascale Computing \u2013 SPPEXA 2013\u20132015, Hans-Joachim Bungartz, Peter Neumann, and Wolfgang E . Nagel (Eds.). Springer International Publishing , Cham , 3\u201323. Peter Bastian, Christian Engwer, Jorrit Fahlke, Markus Geveler, Dominik G\u00f6ddeke, Oleg Iliev, Olaf Ippisch, Ren\u00e9 Milk, Jan Mohring, Steffen M\u00fcthing, Mario Ohlberger, Dirk Ribbrock, and Stefan Turek. 2016. Hardware-based efficiency advances in the EXA-DUNE Project. In Software for Exascale Computing \u2013 SPPEXA 2013\u20132015, Hans-Joachim Bungartz, Peter Neumann, and Wolfgang E. Nagel (Eds.). Springer International Publishing, Cham, 3\u201323."},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.5194\/gmd-9-3803-2016"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1137\/100791634"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.6336"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-010-0110-3"},{"key":"e_1_2_1_13_1","volume-title":"Mund","author":"Deville Michel O.","year":"2002","unstructured":"Michel O. Deville , Paul F. Fischer , and Ernest H . Mund . 2002 . High-Order Methods for Incompressible Fluid Flow. Vol. 9 . Cambridge University Press , Cambridge. Michel O. Deville, Paul F. Fischer, and Ernest H. Mund. 2002. High-Order Methods for Incompressible Fluid Flow. Vol. 9. Cambridge University Press, Cambridge."},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109538"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0010-4655(02)00694-X"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(82)90191-6"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1137\/16M1060017"},{"key":"e_1_2_1_18_1","volume-title":"Discontinuous Galerkin schemes for a class of hamiltonian evolution equations with applications to plasma fluid and kinetic problems. arXiv","author":"Hakim Ammar","year":"1908","unstructured":"Ammar Hakim , Greg Hammett , Eric L. Shi , and Noah Mandell . 2019. Discontinuous Galerkin schemes for a class of hamiltonian evolution equations with applications to plasma fluid and kinetic problems. arXiv 1908 .01814 (2019). Ammar Hakim, Greg Hammett, Eric L. Shi, and Noah Mandell. 2019. Discontinuous Galerkin schemes for a class of hamiltonian evolution equations with applications to plasma fluid and kinetic problems. arXiv 1908.01814 (2019)."},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/3468428"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/1089014.1089021"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.5555\/1557392"},{"key":"e_1_2_1_22_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2017.10.009"},{"key":"e_1_2_1_23_1","volume-title":"METIS: A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices.","author":"Karypis George","year":"1998","unstructured":"George Karypis and Vipin Kumar . 1998 . METIS: A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices. George Karypis and Vipin Kumar. 1998. METIS: A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices."},{"key":"e_1_2_1_24_1","doi-asserted-by":"publisher","DOI":"10.1145\/3424144"},{"key":"e_1_2_1_25_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(99)00141-5"},{"key":"e_1_2_1_26_1","doi-asserted-by":"publisher","DOI":"10.1137\/130928650"},{"key":"e_1_2_1_27_1","doi-asserted-by":"publisher","DOI":"10.1177\/1094342019834644"},{"key":"e_1_2_1_28_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2017.07.039"},{"key":"e_1_2_1_29_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compfluid.2012.04.012"},{"key":"e_1_2_1_30_1","doi-asserted-by":"publisher","DOI":"10.1145\/3325864"},{"key":"e_1_2_1_31_1","volume-title":"A Hermite-like basis for faster matrix-free evaluation of interior penalty discontinuous Galerkin operators. arXiv preprint arXiv:1907.08492","author":"Kronbichler Martin","year":"2019","unstructured":"Martin Kronbichler , Katharina Kormann , Niklas Fehn , Peter Munch , and Julius Witte . 2019. A Hermite-like basis for faster matrix-free evaluation of interior penalty discontinuous Galerkin operators. arXiv preprint arXiv:1907.08492 ( 2019 ). Martin Kronbichler, Katharina Kormann, Niklas Fehn, Peter Munch, and Julius Witte. 2019. A Hermite-like basis for faster matrix-free evaluation of interior penalty discontinuous Galerkin operators. arXiv preprint arXiv:1907.08492 (2019)."},{"key":"e_1_2_1_32_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.5137"},{"key":"e_1_2_1_33_1","doi-asserted-by":"publisher","DOI":"10.1137\/16M110455X"},{"key":"e_1_2_1_34_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(00)00322-4"},{"key":"e_1_2_1_35_1","volume-title":"High-performance implementation of matrix-free high-order discontinuous Galerkin methods. arXiv preprint arXiv:1711.10885","author":"M\u00fcthing Steffen","year":"2017","unstructured":"Steffen M\u00fcthing , Marian Piatkowski , and Peter Bastian . 2017. High-performance implementation of matrix-free high-order discontinuous Galerkin methods. arXiv preprint arXiv:1711.10885 ( 2017 ). Steffen M\u00fcthing, Marian Piatkowski, and Peter Bastian. 2017. High-performance implementation of matrix-free high-order discontinuous Galerkin methods. arXiv preprint arXiv:1711.10885 (2017)."},{"key":"e_1_2_1_36_1","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(80)90005-4"},{"key":"e_1_2_1_37_1","doi-asserted-by":"publisher","DOI":"10.1145\/3054946"},{"key":"e_1_2_1_38_1","doi-asserted-by":"publisher","DOI":"10.1109\/ICPPW.2014.34"},{"key":"e_1_2_1_39_1","doi-asserted-by":"publisher","DOI":"10.1137\/18M1185399"},{"key":"e_1_2_1_40_1","doi-asserted-by":"publisher","DOI":"10.1177\/1094342020945005"},{"key":"e_1_2_1_41_1","doi-asserted-by":"publisher","DOI":"10.1109\/ICPPW.2010.38"},{"key":"e_1_2_1_42_1","doi-asserted-by":"publisher","DOI":"10.1109\/TPS.2012.2188141"},{"key":"e_1_2_1_43_1","doi-asserted-by":"publisher","DOI":"10.5555\/899248"},{"key":"e_1_2_1_44_1","doi-asserted-by":"publisher","DOI":"10.1145\/3319797"},{"key":"e_1_2_1_45_1","doi-asserted-by":"publisher","DOI":"10.1145\/1498765.1498785"}],"container-title":["ACM Transactions on Mathematical Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3469720","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3469720","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T20:17:30Z","timestamp":1750191450000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3469720"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,28]]},"references-count":45,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,12,31]]}},"alternative-id":["10.1145\/3469720"],"URL":"https:\/\/doi.org\/10.1145\/3469720","relation":{},"ISSN":["0098-3500","1557-7295"],"issn-type":[{"value":"0098-3500","type":"print"},{"value":"1557-7295","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,28]]},"assertion":[{"value":"2020-02-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2021-06-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2021-09-28","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}