{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T16:49:46Z","timestamp":1758818986913,"version":"3.40.5"},"reference-count":21,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T00:00:00Z","timestamp":1665014400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We present quadrature schemes to calculate matrices where the so-called modified Hilbert transformation is involved.\nThese matrices occur as temporal parts of Galerkin finite element discretizations of parabolic or hyperbolic problems when the modified Hilbert transformation is used for the variational setting.\nThis work provides the calculation of these matrices to machine precision for arbitrary polynomial degrees and non-uniform meshes.\nThe proposed quadrature schemes are based on weakly singular integral representations of the modified Hilbert transformation.\nFirst, these weakly singular integral representations of the modified Hilbert transformation are proven.\nSecond, using these integral representations, we derive quadrature schemes, which treat the occurring singularities appropriately.\nThus, exponential convergence with respect to the number of quadrature nodes for the proposed quadrature schemes is achieved.\nNumerical results, where this exponential convergence is observed, conclude this work.<\/jats:p>","DOI":"10.1515\/cmam-2022-0150","type":"journal-article","created":{"date-parts":[[2022,10,5]],"date-time":"2022-10-05T17:06:10Z","timestamp":1664989570000},"page":"473-489","source":"Crossref","is-referenced-by-count":6,"title":["Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation"],"prefix":"10.1515","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4146-1474","authenticated-orcid":false,"given":"Marco","family":"Zank","sequence":"first","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik , Universit\u00e4t Wien , Oskar-Morgenstern-Platz 1, 1090 Wien , Austria"}]}],"member":"374","published-online":{"date-parts":[[2022,10,6]]},"reference":[{"key":"2023033113095675538_j_cmam-2022-0150_ref_001","doi-asserted-by":"crossref","unstructured":"W. Bangerth, M. Geiger and R. Rannacher,\nAdaptive Galerkin finite element methods for the wave equation,\nComput. Methods Appl. Math. 10 (2010), no. 1, 3\u201348.","DOI":"10.2478\/cmam-2010-0001"},{"key":"2023033113095675538_j_cmam-2022-0150_ref_002","doi-asserted-by":"crossref","unstructured":"A. Ern and J.-L. Guermond,\nFinite Elements I\u2014Approximation and interpolation,\nTexts Appl. Math. 72,\nSpringer, Cham, 2021.","DOI":"10.1007\/978-3-030-56341-7"},{"key":"2023033113095675538_j_cmam-2022-0150_ref_003","doi-asserted-by":"crossref","unstructured":"W. Gautschi,\nNumerical integration over the square in the presence of algebraic\/logarithmic singularities with an application to aerodynamics,\nNumer. Algorithms 61 (2012), no. 2, 275\u2013290.","DOI":"10.1007\/s11075-012-9611-9"},{"key":"2023033113095675538_j_cmam-2022-0150_ref_004","doi-asserted-by":"crossref","unstructured":"W. Gautschi,\nErratum to: \u201cNumerical integration over the square in the presence of algebraic\/logarithmic singularities with an application to aerodynamics\u201d,\nNumer. Algorithms 64 (2013), no. 4, 759\u2013759.","DOI":"10.1007\/s11075-013-9787-7"},{"key":"2023033113095675538_j_cmam-2022-0150_ref_005","doi-asserted-by":"crossref","unstructured":"O. A. Ladyzhenskaya,\nThe Boundary Value Problems of Mathematical Physics,\nAppl. Math. Sci. 49,\nSpringer, New York, 1985.","DOI":"10.1007\/978-1-4757-4317-3"},{"key":"2023033113095675538_j_cmam-2022-0150_ref_006","doi-asserted-by":"crossref","unstructured":"U. Langer and O. Steinbach,\nSpace-Time Methods\u2014Applications to Partial Differential Equations,\nRadon Ser. Comput. Appl. Math.25,\nDe Gruyter, Berlin, 2019.","DOI":"10.1515\/9783110548488"},{"key":"2023033113095675538_j_cmam-2022-0150_ref_007","doi-asserted-by":"crossref","unstructured":"U. Langer and M. Zank,\nEfficient direct space-time finite element solvers for parabolic initial-boundary value problems in anisotropic Sobolev spaces,\nSIAM J. Sci. Comput. 43 (2021), no. 4, A2714\u2013A2736.","DOI":"10.1137\/20M1358128"},{"key":"2023033113095675538_j_cmam-2022-0150_ref_008","unstructured":"J.-L. Lions and E. Magenes,\nProbl\u00e8mes aux limites non homog\u00e8nes et applications. Vol. 1,\nTrav. Rech. Math. 17,\nDunod, Paris, 1968."},{"key":"2023033113095675538_j_cmam-2022-0150_ref_009","doi-asserted-by":"crossref","unstructured":"R. L\u00f6scher, O. Steinbach and M. Zank,\nNumerical results for an unconditionally stable space-time finite element method for the wave equation. Vol. 26,\nDomain Decomposition Methods in Science and Engineering,\nLect. Notes Comput. Sci. Eng. 145,\nSpringer, Cham, (2022), 587\u2013594.","DOI":"10.1007\/978-3-030-95025-5_68"},{"key":"2023033113095675538_j_cmam-2022-0150_ref_010","doi-asserted-by":"crossref","unstructured":"R. L\u00f6scher, O. Steinbach and M. Zank,\nAn unconditionally stable space-time finite element method for the wave equation, in preparation, 2022.","DOI":"10.1007\/978-3-030-95025-5_68"},{"key":"2023033113095675538_j_cmam-2022-0150_ref_011","unstructured":"J. Ne\u010das,\nLes m\u00e9thodes directes en th\u00e9orie des \u00e9quations elliptiques,\nMasson et Cie, Paris, 1967."},{"key":"2023033113095675538_j_cmam-2022-0150_ref_012","unstructured":"I. Perugia, C. Schwab and M. Zank,\nExponential convergence of \n                  \n                     \n                        \n                           h\n                           \u2062\n                           p\n                        \n                     \n                     \n                     hp\n                  \n               -time-stepping in space-time discretizations of parabolic PDEs, preprint (2022), https:\/\/arxiv.org\/abs\/2203.11879; to appear in ESAIM Math. Model. Numer. Anal."},{"key":"2023033113095675538_j_cmam-2022-0150_ref_013","unstructured":"C. Schwab,\n\ud835\udc5d- and \n                  \n                     \n                        \n                           \n                              h\n                              \u2062\n                              p\n                           \n                        \n                        \n                        hp\n                     \n                  \n               -Finite Element Methods,\nNumer. Math. Sci. Comput.,\nThe Clarendon, New York, 1998."},{"key":"2023033113095675538_j_cmam-2022-0150_ref_014","doi-asserted-by":"crossref","unstructured":"O. Steinbach and A. Missoni,\nA note on a modified Hilbert transform,\nAppl. Anal. (2022), 10.1080\/00036811.2022.2030725.","DOI":"10.1080\/00036811.2022.2030725"},{"key":"2023033113095675538_j_cmam-2022-0150_ref_015","doi-asserted-by":"crossref","unstructured":"O. Steinbach and M. Zank,\nCoercive space-time finite element methods for initial boundary value problems,\nElectron. Trans. Numer. Anal. 52 (2020), 154\u2013194.","DOI":"10.1553\/etna_vol52s154"},{"key":"2023033113095675538_j_cmam-2022-0150_ref_016","unstructured":"O. Steinbach and M. Zank,\nA note on the efficient evaluation of a modified Hilbert transformation,\nJ. Numer. Math. 29 (2021), no. 1, 47\u201361."},{"key":"2023033113095675538_j_cmam-2022-0150_ref_017","unstructured":"V. Thom\u00e9e,\nGalerkin Finite Element Methods for Parabolic Problems, 2nd ed.,\nSpringer Ser. Comput. Math. 25,\nSpringer, Berlin, 2006."},{"key":"2023033113095675538_j_cmam-2022-0150_ref_018","unstructured":"M. Zank,\nInf-Sup Stable Space-Time Methods for Time-Dependent Partial Differential Equations,\nMonographic Series TU Graz 36,\nTU Graz, Graz, 2020."},{"key":"2023033113095675538_j_cmam-2022-0150_ref_019","doi-asserted-by":"crossref","unstructured":"M. Zank,\nAn exact realization of a modified Hilbert transformation for space-time methods for parabolic evolution equations,\nComput. Methods Appl. Math. 21 (2021), no. 2, 479\u2013496.","DOI":"10.1515\/cmam-2020-0026"},{"key":"2023033113095675538_j_cmam-2022-0150_ref_020","unstructured":"M. Zank,\nEfficient direct space-time finite element solvers for the wave equation in second-order formulation, in preparation, 2022."},{"key":"2023033113095675538_j_cmam-2022-0150_ref_021","unstructured":"M. Zank,\nHigh-order discretisations and efficient direct space-time finite element solvers for parabolic initial-boundary value problems,\nto appear in Proceedings of Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0150\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0150\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T17:40:23Z","timestamp":1680284423000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0150\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,6]]},"references-count":21,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,10,6]]},"published-print":{"date-parts":[[2023,4,1]]}},"alternative-id":["10.1515\/cmam-2022-0150"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0150","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2022,10,6]]}}}