{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T06:34:27Z","timestamp":1762324467565,"version":"3.40.5"},"reference-count":26,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-12-1-0484"],"award-info":[{"award-number":["FA9550-12-1-0484"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,7,1]]},"abstract":"Abstract<\/jats:title>\n Numerical integration of the stiffness matrix in higher-order finite element (FE) methods is recognized as one of the heaviest computational tasks in an FE solver.\nThe problem becomes even more relevant when computing the Gram matrix in the algorithm of the Discontinuous Petrov Galerkin (DPG) FE methodology.\nMaking use of 3D tensor-product shape functions, and the concept of sum factorization, known from standard high-order FE and spectral methods, here we take advantage of this idea for the entire exact sequence of FE spaces defined on the hexahedron.\nThe key piece to the presented algorithms is the exact sequence for the one-dimensional element, and use of hierarchical shape functions.\nConsistent with existing results, the presented algorithms for the integration of \n \n \n \n H<\/m:mi>\n 1<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {H^{1}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n \n \n H<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n curl<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {H(\\operatorname{curl})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n \n \n H<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n div<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {H(\\operatorname{div})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, and \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {L^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> inner products, have the \n \n \n \n \ud835\udcaa<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n p<\/m:mi>\n 7<\/m:mn>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{O}(p^{7})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> computational complexity in contrast to the \n \n \n \n \ud835\udcaa<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n p<\/m:mi>\n 9<\/m:mn>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{O}(p^{9})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> cost of conventional integration routines.\nUse of Legendre polynomials for shape functions is critical in this implementation.\nThree boundary value problems under different variational formulations, requiring combinations of \n \n \n \n H<\/m:mi>\n 1<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {H^{1}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n \n \n H<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n div<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {H(\\operatorname{div})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n \n \n H<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n curl<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {H(\\operatorname{curl})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> test shape functions, were chosen to experimentally assess the computation time for constructing DPG element matrices, showing good correspondence with the expected rates.<\/jats:p>","DOI":"10.1515\/cmam-2018-0205","type":"journal-article","created":{"date-parts":[[2019,4,9]],"date-time":"2019-04-09T16:56:03Z","timestamp":1554828963000},"page":"523-555","source":"Crossref","is-referenced-by-count":8,"title":["Fast Integration of DPG Matrices Based on Sum Factorization for all the Energy Spaces"],"prefix":"10.1515","volume":"19","author":[{"given":"Jaime","family":"Mora","sequence":"first","affiliation":[{"name":"Institute for Computational Engineering and Sciences (ICES) , The University of Texas at Austin , 201 E 24th St , Austin , TX 78712 , USA"}]},{"given":"Leszek","family":"Demkowicz","sequence":"additional","affiliation":[{"name":"Institute for Computational Engineering and Sciences (ICES) , The University of Texas at Austin , 201 E 24th St , Austin , TX 78712 , USA"}]}],"member":"374","published-online":{"date-parts":[[2019,4,9]]},"reference":[{"key":"2025051309555085764_j_cmam-2018-0205_ref_001_w2aab3b7e2788b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"M. Ainsworth, G. Andriamaro and O. Davydov,\nBernstein\u2013B\u00e9zier finite elements of arbitrary order and optimal assembly procedures,\nSIAM J. Sci. Comput. 33 (2011), no. 6, 3087\u20133109.","DOI":"10.1137\/11082539X"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_002_w2aab3b7e2788b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"P. Antolin, A. Buffa, F. Calabr\u00f2, M. Martinelli and G. Sangalli,\nEfficient matrix computation for tensor-product isogeometric analysis: The use of sum factorization,\nComput. Methods Appl. Mech. Engrg. 285 (2015), 817\u2013828.","DOI":"10.1016\/j.cma.2014.12.013"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_003_w2aab3b7e2788b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"C. Carstensen, L. Demkowicz and J. Gopalakrishnan,\nA posteriori error control for DPG methods,\nSIAM J. Numer. Anal. 52 (2014), no. 3, 1335\u20131353.","DOI":"10.1137\/130924913"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_004_w2aab3b7e2788b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"C. Carstensen, L. Demkowicz and J. Gopalakrishnan,\nBreaking spaces and forms for the DPG method and applications including Maxwell equations,\nComput. Math. Appl. 72 (2016), no. 3, 494\u2013522.","DOI":"10.1016\/j.camwa.2016.05.004"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_005_w2aab3b7e2788b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"L. Demkowicz and J. Gopalakrishnan,\nA class of discontinuous Petrov\u2013Galerkin methods. II. Optimal test functions,\nNumer. Methods Partial Differential Equations 27 (2011), no. 1, 70\u2013105.","DOI":"10.1002\/num.20640"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_006_w2aab3b7e2788b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"L. Demkowicz and J. Gopalakrishnan,\nAn overview of the discontinuous Petrov Galerkin method,\nRecent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations,\nIMA Vol. Math. Appl. 157,\nSpringer, Cham (2014), 149\u2013180.","DOI":"10.1007\/978-3-319-01818-8_6"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_007_w2aab3b7e2788b1b6b1ab2b1b7Aa","unstructured":"L. Demkowicz and J. Gopalakrishnan,\nDiscontinuous Petrov\u2013Galerkin (DPG) method,\nICES Report 15-20, The University of Texas at Austin, 2015."},{"key":"2025051309555085764_j_cmam-2018-0205_ref_008_w2aab3b7e2788b1b6b1ab2b1b8Aa","doi-asserted-by":"crossref","unstructured":"L. Demkowicz, J. Gopalakrishnan, S. Nagaraj and P. Sep\u00falveda,\nA spacetime DPG method for the Schr\u00f6dinger equation,\nSIAM J. Numer. Anal. 55 (2017), no. 4, 1740\u20131759.","DOI":"10.1137\/16M1099765"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_009_w2aab3b7e2788b1b6b1ab2b1b9Aa","doi-asserted-by":"crossref","unstructured":"L. Demkowicz, J. Gopalakrishnan and A. H. Niemi,\nA class of discontinuous Petrov\u2013Galerkin methods. Part III: Adaptivity,\nAppl. Numer. Math. 62 (2012), no. 4, 396\u2013427.","DOI":"10.1016\/j.apnum.2011.09.002"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_010_w2aab3b7e2788b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"L. Demkowicz, J. Kurtz, D. Pardo, M. Paszy\u0144ski, W. Rachowicz and A. Zdunek,\nComputing with hp Finite Elements. II. Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications,\nChapman & Hall\/CRC, New York, 2007.","DOI":"10.1201\/9781420011692"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_011_w2aab3b7e2788b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"F. Fuentes, L. Demkowicz and A. Wilder,\nUsing a DPG method to validate DMA experimental calibration of viscoelastic materials,\nComput. Methods Appl. Mech. Engrg. 325 (2017), 748\u2013765.","DOI":"10.1016\/j.cma.2017.07.012"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_012_w2aab3b7e2788b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"F. Fuentes, B. Keith, L. Demkowicz and S. Nagaraj,\nOrientation embedded high order shape functions for the exact sequence elements of all shapes,\nComput. Math. Appl. 70 (2015), no. 4, 353\u2013458.","DOI":"10.1016\/j.camwa.2015.04.027"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_013_w2aab3b7e2788b1b6b1ab2b1c13Aa","unstructured":"F. Hellwig,\nThree low-order dPG methods for linear elasticity,\nMaster\u2019s thesis, Humboldt-Universit\u00e4t zu Berlin, Berlin, 2014."},{"key":"2025051309555085764_j_cmam-2018-0205_ref_014_w2aab3b7e2788b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"G. E. Karniadakis and S. J. Sherwin,\nSpectral\/hp Element Methods for Computational Fluid Dynamics, 2nd ed.,\nNumer. Math. Sci. Comput.,\nOxford University, New York, 2005.","DOI":"10.1093\/acprof:oso\/9780198528692.001.0001"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_015_w2aab3b7e2788b1b6b1ab2b1c15Aa","doi-asserted-by":"crossref","unstructured":"B. Keith, F. Fuentes and L. Demkowicz,\nThe DPG methodology applied to different variational formulations of linear elasticity,\nComput. Methods Appl. Mech. Engrg. 309 (2016), 579\u2013609.","DOI":"10.1016\/j.cma.2016.05.034"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_016_w2aab3b7e2788b1b6b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"B. Keith, P. Knechtges, N. V. Roberts, S. Elgeti, M. Behr and L. Demkowicz,\nAn ultraweak DPG method for viscoelastic fluids,\nJ. Non-Newton. Fluid Mech. 247 (2017), 107\u2013122.","DOI":"10.1016\/j.jnnfm.2017.06.006"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_017_w2aab3b7e2788b1b6b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"B. Keith, S. Petrides, F. Fuentes and L. Demkowicz,\nDiscrete least-squares finite element methods,\nComput. Methods Appl. Mech. Engrg. 327 (2017), 226\u2013255.","DOI":"10.1016\/j.cma.2017.08.043"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_018_w2aab3b7e2788b1b6b1ab2b1c18Aa","unstructured":"J. P. Kurtz,\nFully Automatic hp-Adaptivity for Acoustic and Electromagnetic Scattering in Three Dimensions,\nProQuest LLC, Ann Arbor, 2007."},{"key":"2025051309555085764_j_cmam-2018-0205_ref_019_w2aab3b7e2788b1b6b1ab2b1c19Aa","doi-asserted-by":"crossref","unstructured":"J. M. Melenk, K. Gerdes and C. Schwab,\nFully discrete hp-finite elements: Fast quadrature,\nComput. Methods Appl. Math. 190 (2001), no. 32, 4339\u20134364.","DOI":"10.1016\/S0045-7825(00)00322-4"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_020_w2aab3b7e2788b1b6b1ab2b1c20Aa","doi-asserted-by":"crossref","unstructured":"S. Nagaraj, J. Grosek, S. Petrides, L. F. Demkowicz and J. Mora,\nA 3D DPG Maxwell approach to nonlinear Raman gain in fiber laser amplifiers,\nJ. Comput. Phys. X 2 (2019), Article ID 100002.","DOI":"10.1016\/j.jcpx.2019.100002"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_021_w2aab3b7e2788b1b6b1ab2b1c21Aa","doi-asserted-by":"crossref","unstructured":"S. Nagaraj, S. Petrides and L. F. Demkowicz,\nConstruction of DPG Fortin operators for second order problems,\nComput. Math. Appl. 74 (2017), no. 8, 1964\u20131980.","DOI":"10.1016\/j.camwa.2017.05.030"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_022_w2aab3b7e2788b1b6b1ab2b1c22Aa","doi-asserted-by":"crossref","unstructured":"J.-C. N\u00e9d\u00e9lec,\nMixed finite elements in \ud835\udc113{\\mathbf{R}}^{3},\nNumer. Math. 35 (1980), no. 3, 315\u2013341.","DOI":"10.1007\/BF01396415"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_023_w2aab3b7e2788b1b6b1ab2b1c23Aa","doi-asserted-by":"crossref","unstructured":"S. A. Orszag,\nSpectral methods for problems in complex geometries,\nJ. Comput. Phys. 37 (1980), no. 1, 70\u201392.","DOI":"10.1016\/0021-9991(80)90005-4"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_024_w2aab3b7e2788b1b6b1ab2b1c24Aa","doi-asserted-by":"crossref","unstructured":"S. Petrides and L. F. Demkowicz,\nAn adaptive DPG method for high frequency time-harmonic wave propagation problems,\nComput. Math. Appl. 74 (2017), no. 8, 1999\u20132017.","DOI":"10.1016\/j.camwa.2017.06.044"},{"key":"2025051309555085764_j_cmam-2018-0205_ref_025_w2aab3b7e2788b1b6b1ab2b1c25Aa","unstructured":"N. V. Roberts,\nA discontinuous Petrov\u2013Galerkin methodology for incompressible flow problems,\nPhD thesis, The University of Texas at Austin, Austin, 2013."},{"key":"2025051309555085764_j_cmam-2018-0205_ref_026_w2aab3b7e2788b1b6b1ab2b1c26Aa","doi-asserted-by":"crossref","unstructured":"A. Vaziri Astaneh, F. Fuentes, J. Mora and L. Demkowicz,\nHigh-order polygonal discontinuous Petrov\u2013Galerkin (PolyDPG) methods using ultraweak formulations,\nComput. Methods Appl. Mech. Engrg. 332 (2018), 686\u2013711.","DOI":"10.1016\/j.cma.2017.12.011"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/19\/3\/article-p523.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2018-0205\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2018-0205\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T09:57:17Z","timestamp":1747130237000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2018-0205\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,9]]},"references-count":26,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,4,9]]},"published-print":{"date-parts":[[2019,7,1]]}},"alternative-id":["10.1515\/cmam-2018-0205"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2018-0205","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"type":"electronic","value":"1609-9389"},{"type":"print","value":"1609-4840"}],"subject":[],"published":{"date-parts":[[2019,4,9]]}}}