{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T11:14:30Z","timestamp":1760267670450},"reference-count":20,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,7,1]]},"abstract":"Abstract<\/jats:title>\n We propose and investigate the application of alternative enriched test spaces in the discontinuous Petrov\u2013Galerkin (DPG) finite element framework for singular perturbation linear problems, with an emphasis on 2D convection-dominated diffusion. Providing robust \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> error estimates for the field variables is considered a convenient feature for this class of problems, since this norm would not account for the large gradients present in boundary layers. With this requirement in mind, Demkowicz and others have previously formulated special test norms, which through DPG deliver the desired \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> convergence. However, robustness has only been verified through numerical experiments for tailored<\/jats:italic> test norms which are problem-specific, whereas the quasi-optimal<\/jats:italic> test norm (not problem specific) has failed such tests due to the difficulty to resolve the optimal test functions sought in the DPG technology. To address this issue (i.e. improve optimal test functions resolution for the quasi-optimal test norm), we propose to discretize the local test spaces with functions that depend on the perturbation parameter \u03f5. Explicitly, we work with B-spline spaces defined on an \u03f5-dependent Shishkin submesh. Two examples are run using adaptive h<\/jats:italic>-refinement to compare the performance of proposed test spaces with that of standard test spaces. We also include a modified norm and a continuation strategy aiming to improve time performance and briefly experiment with these ideas.<\/jats:p>","DOI":"10.1515\/cmam-2018-0207","type":"journal-article","created":{"date-parts":[[2019,7,3]],"date-time":"2019-07-03T09:40:31Z","timestamp":1562146831000},"page":"603-630","source":"Crossref","is-referenced-by-count":6,"title":["Alternative Enriched Test Spaces in the DPG Method for Singular Perturbation Problems"],"prefix":"10.1515","volume":"19","author":[{"given":"Jacob","family":"Salazar","sequence":"first","affiliation":[{"name":"Institute for Computational Engineering and Sciences (ICES) , The University of Texas at Austin , 2 01 E 24th St , Austin , TX 78712 , USA"}]},{"given":"Jaime","family":"Mora","sequence":"additional","affiliation":[{"name":"Institute for Computational Engineering and Sciences (ICES) , The University of Texas at Austin , 2 01 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 , 2 01 E 24th St , Austin , TX 78712 , USA"}]}],"member":"374","published-online":{"date-parts":[[2019,4,11]]},"reference":[{"key":"2023033110340744728_j_cmam-2018-0207_ref_001_w2aab3b7e3324b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"A. Buffa, J. Rivas, G. Sangalli and R. V\u00e1zquez,\nIsogeometric discrete differential forms in three dimensions,\nSIAM J. Numer. Anal. 49 (2011), no. 2, 818\u2013844.","DOI":"10.1137\/100786708"},{"key":"2023033110340744728_j_cmam-2018-0207_ref_002_w2aab3b7e3324b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"D. Broersen, W. Dahmen and R. P. Stevenson,\nOn the stability of DPG formulattions of transport equations,\nMath. Comp. 87 (2018), no. 311, 1051\u20131082.","DOI":"10.1090\/mcom\/3242"},{"key":"2023033110340744728_j_cmam-2018-0207_ref_003_w2aab3b7e3324b1b6b1ab2ab3Aa","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":"2023033110340744728_j_cmam-2018-0207_ref_004_w2aab3b7e3324b1b6b1ab2ab4Aa","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":"2023033110340744728_j_cmam-2018-0207_ref_005_w2aab3b7e3324b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"J. A. Cottrell, T. J. R. Hughes and Y. Bazilevs,\nIsogeometric Analysis,\nJohn Wiley & Sons, Chichester, 2009.","DOI":"10.1002\/9780470749081"},{"key":"2023033110340744728_j_cmam-2018-0207_ref_006_w2aab3b7e3324b1b6b1ab2ab6Aa","unstructured":"W. Dahmen and R. P. Stevenson,\nAdaptive strategies for transport equations,\npreprint (2018), https:\/\/arxiv.org\/abs\/1809.02055v1."},{"key":"2023033110340744728_j_cmam-2018-0207_ref_007_w2aab3b7e3324b1b6b1ab2ab7Aa","unstructured":"L. Demkowicz,\nComputing with hp Finite Elements. I: One and Two Dimensional Elliptic and Maxwell Problems,\nChapman & Hall\/CRC Press, New York, 2006."},{"key":"2023033110340744728_j_cmam-2018-0207_ref_008_w2aab3b7e3324b1b6b1ab2ab8Aa","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":"2023033110340744728_j_cmam-2018-0207_ref_009_w2aab3b7e3324b1b6b1ab2ab9Aa","unstructured":"L. Demkowicz and J. Gopalakrishnan,\nDiscontinuous Petrov\u2013Galerkin (DPG) method,\nICES Report 15-20, The University of Texas at Austin, 2015."},{"key":"2023033110340744728_j_cmam-2018-0207_ref_010_w2aab3b7e3324b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"L. Demkowicz, J. Gopalakrishnan and A. H. Niemi,\nA class of discontinuous Petrov-Galerkin methods. Part III: Adaptivity,\nAppl. Numer. Math. 62 (2012), no. 4, 396\u2013427.","DOI":"10.1016\/j.apnum.2011.09.002"},{"key":"2023033110340744728_j_cmam-2018-0207_ref_011_w2aab3b7e3324b1b6b1ab2ac11Aa","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":"2023033110340744728_j_cmam-2018-0207_ref_012_w2aab3b7e3324b1b6b1ab2ac12Aa","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":"2023033110340744728_j_cmam-2018-0207_ref_013_w2aab3b7e3324b1b6b1ab2ac13Aa","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":"2023033110340744728_j_cmam-2018-0207_ref_014_w2aab3b7e3324b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"M. K. Kadalbajoo and D. Kumar,\nFitted mesh B-spline collocation method for singularly perturbed differential-difference equations with small delay,\nAppl. Math. 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Oden and L. Demkowicz,\nApplied Functional Analysis,\nChapman and Hall\/CRC, New York, 2017."},{"key":"2023033110340744728_j_cmam-2018-0207_ref_018_w2aab3b7e3324b1b6b1ab2ac18Aa","doi-asserted-by":"crossref","unstructured":"M. E. Rognes, R. C. Kirby and A. Logg,\nEfficient assembly of H\u2062(div){H({\\rm div})} and H\u2062(curl){H({\\rm curl})} conforming finite elements,\nSIAM J. Sci. Comput. 31 (2009\/10), no. 6, 4130\u20134151.","DOI":"10.1137\/08073901X"},{"key":"2023033110340744728_j_cmam-2018-0207_ref_019_w2aab3b7e3324b1b6b1ab2ac19Aa","unstructured":"G. I. Shishkin,\nGrid approximation of singularly perturbed elliptic and parabolic equations,\nUral Br. Russ. Acad. Sci. Ekaterinburg 727 (1992), 728\u2013729."},{"key":"2023033110340744728_j_cmam-2018-0207_ref_020_w2aab3b7e3324b1b6b1ab2ac20Aa","doi-asserted-by":"crossref","unstructured":"J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo and V. M. Calo,\nA class of discontinuous Petrov-Galerkin methods. 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