{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T10:25:52Z","timestamp":1762251952862},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2013,6,10]],"date-time":"2013-06-10T00:00:00Z","timestamp":1370822400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2013,12]]},"DOI":"10.1007\/s10915-013-9726-8","type":"journal-article","created":{"date-parts":[[2013,6,9]],"date-time":"2013-06-09T08:26:39Z","timestamp":1370766399000},"page":"536-581","source":"Crossref","is-referenced-by-count":71,"title":["General DG-Methods for Highly Indefinite Helmholtz Problems"],"prefix":"10.1007","volume":"57","author":[{"given":"J. M.","family":"Melenk","sequence":"first","affiliation":[]},{"given":"A.","family":"Parsania","sequence":"additional","affiliation":[]},{"given":"S.","family":"Sauter","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2013,6,10]]},"reference":[{"key":"9726_CR1","volume-title":"Sobolev Spaces","author":"RA Adams","year":"1975","unstructured":"Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)"},{"issue":"1\u20133","key":"9726_CR2","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s10915-005-9044-x","volume":"27","author":"M Ainsworth","year":"2006","unstructured":"Ainsworth, M., Monk, P., Muniz, W.: Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. J. Sci. Comput. 27(1\u20133), 5\u201340 (2006)","journal-title":"J. Sci. Comput."},{"issue":"2","key":"9726_CR3","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1137\/S0036142903423460","volume":"42","author":"M Ainsworth","year":"2004","unstructured":"Ainsworth, M.: Discrete dispersion relation for $$hp$$ hp -version finite element approximation at high wave number. SIAM J. Numer. Anal. 42(2), 553\u2013575 (2004)","journal-title":"SIAM J. Numer. Anal."},{"issue":"5","key":"9726_CR4","doi-asserted-by":"crossref","first-page":"3910","DOI":"10.1137\/080724976","volume":"47","author":"M Ainsworth","year":"2009","unstructured":"Ainsworth, M., Wajid, H.: Dispersive and dissipative behavior of the spectral element method. SIAM J. Numer. Anal. 47(5), 3910\u20133937 (2009)","journal-title":"SIAM J. Numer. Anal."},{"issue":"18","key":"9726_CR5","doi-asserted-by":"crossref","first-page":"3443","DOI":"10.1002\/(SICI)1097-0207(19970930)40:18<3443::AID-NME221>3.0.CO;2-1","volume":"40","author":"I Babu\u0161ka","year":"1997","unstructured":"Babu\u0161ka, I., Ihlenburg, F., Strouboulis, T., Gangaraj, S.K.: A posteriori error estimation for finite element solutions of Helmholtz\u2019 equation. I. The quality of local indicators and estimators. Int. J. Numer. Methods Eng. 40(18), 3443\u20133462 (1997)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"9726_CR6","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1090\/S0025-5718-1978-0501990-5","volume":"32","author":"J Bramble","year":"1978","unstructured":"Bramble, J., Scott, R.: Simultaneous approximation in scales of Banach spaces. Math. Comput. 32, 947\u2013954 (1978)","journal-title":"Math. Comput."},{"key":"9726_CR7","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1051\/m2an:2008033","volume":"42","author":"A Buffa","year":"2008","unstructured":"Buffa, A., Monk, P.: Error estimates for the Ultra Weak Variational Formulation of the Helmholtz Equation. Math. Model. Numer. Anal. 42, 925\u2013940 (2008)","journal-title":"Math. Model. Numer. Anal."},{"key":"9726_CR8","doi-asserted-by":"crossref","unstructured":"Cessenat, O., Despr\u00e9s, B.: Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation. SIAM J. Numer. Anal. 35, 255\u2013299, 1594\u20131607 (1998)","DOI":"10.1137\/S0036142995285873"},{"key":"9726_CR9","unstructured":"Cessenat, O., Despr\u00e9s, B.: Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation. J. Comput. Acoust. 11, 227\u2013238 (2003)"},{"issue":"1","key":"9726_CR10","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1142\/S021820250600108X","volume":"16","author":"P Cummings","year":"2006","unstructured":"Cummings, P., Feng, X.: Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations. Math. Models Methods Appl. Sci. 16(1), 139\u2013160 (2006)","journal-title":"Math. Models Methods Appl. Sci."},{"key":"9726_CR11","volume-title":"Mixed Finite Elements, Compatibility Conditions, and Applications. Lectures Notes in Mathematics","author":"L Demkowicz","year":"2008","unstructured":"Demkowicz, L.: Polynomial exact sequences and projection-based interpolation with applications to Maxwell\u2019s equations. In: Boffi, D., Brezzi, F., Demkowicz, L., Dur\u00e1n, L.F., Falk, R., Fortin, M. (eds.) Mixed Finite Elements, Compatibility Conditions, and Applications. Lectures Notes in Mathematics, vol. 1939. Springer, Berlin (2008)"},{"key":"9726_CR12","doi-asserted-by":"crossref","unstructured":"Demkowicz, L., Kurtz, J., Pardo, D., Paszy\u0144ski, M., Rachowicz, W., Zdunek, A.: Computing with $$hp$$ hp -adaptive finite elements. In: Chapman & Hall\/CRC Applied Mathematics and Nonlinear Science Series, Vol. 2. Chapman & Hall\/CRC, Boca Raton, FL. Frontiers: three dimensional elliptic and Maxwell problems with applications (2008)","DOI":"10.1201\/9781420011692"},{"key":"9726_CR13","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1002\/(SICI)1097-0207(19991010)46:4<471::AID-NME684>3.0.CO;2-6","volume":"46","author":"A Deraemaeker","year":"1999","unstructured":"Deraemaeker, A., Babu\u0161ka, I., Bouillard, P.: Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. Int. J. Numer. Meth. Eng. 46, 471\u2013499 (1999)","journal-title":"Int. J. Numer. Meth. Eng."},{"key":"9726_CR14","first-page":"939","volume":"318","author":"B Despr\u00e9s","year":"1994","unstructured":"Despr\u00e9s, B.: Sur une formulation variationelle de type ultra-faible. C.R. Acad. Sci. Paris Ser. I 318, 939\u2013944 (1994)","journal-title":"C.R. Acad. Sci. Paris Ser. I"},{"key":"9726_CR15","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1007\/978-3-642-22061-6_9","volume-title":"Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering","author":"S Esterhazy","year":"2012","unstructured":"Esterhazy, S., Melenk, J.M.: On stability of discretizations of the Helmholtz equation. In: Graham, I.G., Hou, T.Y., Lakkis, O., Scheichl, R. (eds.) Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering, vol. 83, pp. 285\u2013324. Springer, Berlin (2012)"},{"key":"9726_CR16","unstructured":"Esterhazy, S., Melenk, J.M.: An analysis of discretizations of the Helmholtz equation in $$L^2$$ L 2 and negative norms. ASC-report 31\/2012. Institute for Analysis and Scientific Computing. Vienna University of Technology (2012)"},{"issue":"2","key":"9726_CR17","doi-asserted-by":"crossref","first-page":"335","DOI":"10.4208\/cicp.081209.070710s","volume":"11","author":"M Grigoroscuta-Strugaru","year":"2012","unstructured":"Grigoroscuta-Strugaru, M., Amara, M., Calandra, H., Djellouli, R.: A modified discontinuous Galerkin method for solving efficiently Helmholtz problems. Commun. Comput. Phys. 11(2), 335\u2013350 (2012)","journal-title":"Commun. Comput. Phys."},{"key":"9726_CR18","volume-title":"Elliptic Problems in Nonsmooth Domains","author":"P Grisvard","year":"1985","unstructured":"Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)"},{"issue":"4","key":"9726_CR19","doi-asserted-by":"crossref","first-page":"2872","DOI":"10.1137\/080737538","volume":"47","author":"X Feng","year":"2009","unstructured":"Feng, X., Wu, H.: Discontinuous Galerkin methods for the Helmholtz equation with large wave number. SIAM J. Numer. Anal. 47(4), 2872\u20132896 (2009)","journal-title":"SIAM J. Numer. Anal."},{"key":"9726_CR20","doi-asserted-by":"crossref","first-page":"1997","DOI":"10.1090\/S0025-5718-2011-02475-0","volume":"80","author":"X Feng","year":"2011","unstructured":"Feng, X., Wu, H.: $$hp$$ hp -discontinuous Galerkin methods for the Helmholtz equation with large wave number. Math. Comput. 80, 1997\u20132024 (2011)","journal-title":"Math. Comput."},{"key":"9726_CR21","unstructured":"Feng, X., Xing, Y.: Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number. Math. Comput. 82, 1269\u20131296 (2013)"},{"issue":"1","key":"9726_CR22","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1007\/s10915-009-9315-z","volume":"42","author":"E Georgoulis","year":"2010","unstructured":"Georgoulis, E., Hall, E., Melenk, J.M.: On the suboptimality of the $$p$$ p -version interior penalty discontinuous galerkin method. J. Sci. Comput. 42(1), 54\u201367 (2010)","journal-title":"J. Sci. Comput."},{"key":"9726_CR23","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1051\/m2an\/2009002","volume":"43","author":"CJ Gittelson","year":"2009","unstructured":"Gittelson, C.J., Hiptmair, R., Perugia, I.: Plane wave discontinuous Galerkin methods: Analysis of the $$h$$ h -version. Math. Model. Numer. Anal. 43, 297\u2013331 (2009)","journal-title":"Math. Model. Numer. Anal."},{"key":"9726_CR24","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1002\/nme.1620070405","volume":"7","author":"WJ Gordon","year":"1973","unstructured":"Gordon, W.J., Hall, ChA: Construction of curvilinear co-ordinate systems and applications to mesh generation. Int. J. Numer. Methods Eng. 7, 461\u2013477 (1973)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"9726_CR25","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/BF01436298","volume":"21","author":"WJ Gordon","year":"1973","unstructured":"Gordon, W.J., Hall, C.A.: Transfinite element methods: blending function interpolation over arbitrary curved element domains. Numer. Math. 21, 109\u2013129 (1973)","journal-title":"Numer. Math."},{"issue":"13\u201316","key":"9726_CR26","doi-asserted-by":"crossref","first-page":"1594","DOI":"10.1016\/j.cma.2005.05.030","volume":"195","author":"I Harari","year":"2006","unstructured":"Harari, I.: A survey of finite element methods for time-harmonic acoustics. Comput. Methods. Appl. Mech. Eng. 195(13\u201316), 1594\u20131607 (2006)","journal-title":"Comput. Methods. Appl. Mech. Eng."},{"issue":"3","key":"9726_CR27","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1016\/0045-7825(92)90006-6","volume":"98","author":"I Harari","year":"1992","unstructured":"Harari, I., Hughes, T.J.R.: Galerkin\/least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains. Comput. Methods. Appl. Mech. Eng. 98(3), 411\u2013454 (1992)","journal-title":"Comput. Methods. Appl. Mech. Eng."},{"key":"9726_CR28","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1137\/090761057","volume":"49","author":"R Hiptmair","year":"2011","unstructured":"Hiptmair, R., Moiola, A., Perugia, I.: Plane Wave Discontinuous Galerkin Methods for $$2D$$ 2 D Helmholtz equation: analysis of the $$p$$ p -version. SIAM J. Numer. Anal. 49, 264\u2013284 (2011)","journal-title":"SIAM J. Numer. Anal."},{"key":"9726_CR29","first-page":"350","volume":"25","author":"T Huttunen","year":"2007","unstructured":"Huttunen, T., Monk, P.: The use of plane waves to approximate wave propagation in anisotropic media. J. Comput. Math. 25, 350\u2013367 (2007)","journal-title":"J. Comput. Math."},{"key":"9726_CR30","doi-asserted-by":"crossref","unstructured":"Ihlenburg, F.: Finite Element Analysis of Acoustic Scattering, vol. 132. Applied Mathematical Sciences. Springer, New York (1998)","DOI":"10.1007\/b98828"},{"issue":"22","key":"9726_CR31","doi-asserted-by":"crossref","first-page":"3745","DOI":"10.1002\/nme.1620382203","volume":"38","author":"F Ihlenburg","year":"1995","unstructured":"Ihlenburg, F., Babu\u0161ka, I.: Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. Int. J. Numer. Methods Eng. 38(22), 3745\u20133774 (1995)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"9726_CR32","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1137\/S0036142994272337","volume":"34","author":"F Ihlenburg","year":"1997","unstructured":"Ihlenburg, F., Babu\u0161ka, I.: Finite element solution of the Helmholtz equation with high wavenumber part II: the h-p-version of the FEM. SIAM J. Numer. Anal. 34, 315\u2013358 (1997)","journal-title":"SIAM J. Numer. Anal."},{"issue":"6","key":"9726_CR33","doi-asserted-by":"crossref","first-page":"2340","DOI":"10.1137\/100786034","volume":"49","author":"M L\u00f6hndorf","year":"2011","unstructured":"L\u00f6hndorf, M., Melenk, J.M.: Wavenumber-explicit $$hp$$ hp -BEM for high frequency scattering. SIAM J. Numer. Anal. 49(6), 2340\u20132363 (2011)","journal-title":"SIAM J. Numer. Anal."},{"key":"9726_CR34","unstructured":"Melenk, J.M.: On Generalized Finite Element Methods. PhD thesis, University of Maryland at College Park (1995)"},{"issue":"4","key":"9726_CR35","doi-asserted-by":"crossref","first-page":"2599","DOI":"10.1137\/100784072","volume":"44","author":"JM Melenk","year":"2012","unstructured":"Melenk, J.M.: Mapping properties of combined field Helmholtz boundary integral operators. SIAM J. Math. Anal. 44(4), 2599\u20132636 (2012)","journal-title":"SIAM J. Math. Anal."},{"key":"9726_CR36","doi-asserted-by":"crossref","first-page":"1871","DOI":"10.1090\/S0025-5718-10-02362-8","volume":"79","author":"JM Melenk","year":"2010","unstructured":"Melenk, J.M., Sauter, S.: Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary condition. Math. Comput. 79, 1871\u20131914 (2010)","journal-title":"Math. Comput."},{"key":"9726_CR37","doi-asserted-by":"crossref","first-page":"1210","DOI":"10.1137\/090776202","volume":"49","author":"JM Melenk","year":"2011","unstructured":"Melenk, J.M., Sauter, S.: Wavenumber explicit convergence analysis for finite element discretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49, 1210\u20131243 (2011)","journal-title":"SIAM J. Numer. Anal."},{"key":"9726_CR38","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/S0045-7825(98)00326-0","volume":"175","author":"P Monk","year":"1999","unstructured":"Monk, P., Wang, D.Q.: A least squares methods for the Helmholtz equation. Comput. Meth. Appl. Mech. Eng. 175, 121\u2013136 (1999)","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"9726_CR39","volume-title":"Asymptotics and Special Functions","author":"FWJ Olver","year":"1974","unstructured":"Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)"},{"key":"9726_CR40","unstructured":"Parsania, A.: Convergence analysis for finite element discretizations of highly indefinite problems. Doctoral thesis, Institut f\u00fcr Mathematik, Universit\u00e4t Z\u00fcrich (2012)"},{"key":"9726_CR41","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1090\/S0025-5718-1974-0373326-0","volume":"28","author":"A Schatz","year":"1974","unstructured":"Schatz, A.: An observation concerning Ritz\u2013Galerkin methods with indefinite bilinear forms. Math. Comput. 28, 959\u2013962 (1974)","journal-title":"Math. Comput."},{"key":"9726_CR42","unstructured":"Schwab, C.: p- and hp-Finite Element Methods. Oxford University Press, New York (1998)"},{"key":"9726_CR43","volume-title":"Interpolation theory, function spaces, differential operators","author":"H Triebel","year":"1995","unstructured":"Triebel, H.: Interpolation theory, function spaces, differential operators, 2nd edn. Johann Ambrosius Barth, Heidelberg (1995)","edition":"2"},{"key":"9726_CR44","unstructured":"Wu, H.: Pre-asymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part I: linear version. Technical report (2011)"},{"key":"9726_CR45","unstructured":"Zhu, L., Wu, H.: Pre-asymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part II: $$hp$$ hp version. Technical report (2009)"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-013-9726-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-013-9726-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-013-9726-8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T14:57:14Z","timestamp":1645714634000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-013-9726-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,6,10]]},"references-count":45,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,12]]}},"alternative-id":["9726"],"URL":"https:\/\/doi.org\/10.1007\/s10915-013-9726-8","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,6,10]]}}}