{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T09:32:05Z","timestamp":1762507925227},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,11,18]],"date-time":"2019-11-18T00:00:00Z","timestamp":1574035200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,11,18]],"date-time":"2019-11-18T00:00:00Z","timestamp":1574035200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"TU Wien"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2020,2]]},"abstract":"Abstract<\/jats:title>For functions$$u\\in H^1(\\Omega )$$<\/jats:tex-math>u<\/mml:mi>\u2208<\/mml:mo>H<\/mml:mi>1<\/mml:mn><\/mml:msup>(<\/mml:mo>\u03a9<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>in a bounded polytope$$\\Omega \\subset {\\mathbb {R}}^d$$<\/jats:tex-math>\u03a9<\/mml:mi>\u2282<\/mml:mo>R<\/mml:mi><\/mml:mrow>d<\/mml:mi><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>$$d=1,2,3$$<\/jats:tex-math>d<\/mml:mi>=<\/mml:mo>1<\/mml:mn>,<\/mml:mo>2<\/mml:mn>,<\/mml:mo>3<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>with plane sides for$$d=2,3$$<\/jats:tex-math>d<\/mml:mi>=<\/mml:mo>2<\/mml:mn>,<\/mml:mo>3<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>which are Gevrey regular in$$\\overline{\\Omega }\\backslash {\\mathscr {S}}$$<\/jats:tex-math>\u03a9<\/mml:mi>\u00af<\/mml:mo><\/mml:mover>\\<\/mml:mo>S<\/mml:mi><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>with point singularities concentrated at a set$${\\mathscr {S}}\\subset \\overline{\\Omega }$$<\/jats:tex-math>S<\/mml:mi>\u2282<\/mml:mo>\u03a9<\/mml:mi>\u00af<\/mml:mo><\/mml:mover><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>consisting of a finite number of points in$$\\overline{\\Omega }$$<\/jats:tex-math>\u03a9<\/mml:mi>\u00af<\/mml:mo><\/mml:mover><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we prove exponential rates of convergence ofhp<\/jats:italic>-version continuous Galerkin finite element methods on affine families of regular, simplicial meshes in$$\\Omega $$<\/jats:tex-math>\u03a9<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The simplicial meshes are geometrically refined towards$${\\mathscr {S}}$$<\/jats:tex-math>S<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>but are otherwise unstructured.<\/jats:p>","DOI":"10.1007\/s00211-019-01085-z","type":"journal-article","created":{"date-parts":[[2019,11,18]],"date-time":"2019-11-18T19:02:41Z","timestamp":1574103761000},"page":"323-346","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Exponential convergence in $$H^1$$ of hp-FEM for Gevrey regularity with isotropic singularities"],"prefix":"10.1007","volume":"144","author":[{"given":"M.","family":"Feischl","sequence":"first","affiliation":[]},{"given":"Ch.","family":"Schwab","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,11,18]]},"reference":[{"key":"1085_CR1","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1016\/j.apnum.2013.12.004","volume":"95","author":"M Aurada","year":"2015","unstructured":"Aurada, M., Feischl, M., F\u00fchrer, T., Karkulik, M., Praetorius, D.: Energy norm based error estimators for adaptive BEM for hypersingular integral equations. Appl. Numer. Math. 95, 15\u201335 (2015)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"1085_CR2","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1137\/0519014","volume":"19","author":"I Babu\u0161ka","year":"1988","unstructured":"Babu\u0161ka, I., Guo, B.Q.: Regularity of the solution of elliptic problems with piecewise analytic data. I. Boundary value problems for linear elliptic equation of second order. SIAM J. Math. Anal. 19(1), 172\u2013203 (1988)","journal-title":"SIAM J. Math. Anal."},{"key":"1085_CR3","doi-asserted-by":"publisher","first-page":"4967","DOI":"10.1016\/j.jcp.2012.04.002","volume":"231","author":"G Bao","year":"2012","unstructured":"Bao, G., Hu, G., Liu, D.: An $$h$$-adaptive finite element solver for the calculation of the electronic structures. J. Comput. Phys. 231, 4967\u20134979 (2012)","journal-title":"J. Comput. Phys."},{"key":"1085_CR4","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1080\/03605308508820383","volume":"10","author":"P Bolley","year":"1985","unstructured":"Bolley, P., Dauge, M., Camus, J.: R\u00e9gularit\u00e9 Gevrey pour le probl\u00e8me de Dirichlet dans des domaines \u00e0 singularit\u00e9s coniques. Commun. Partial Differ. Equ. 10, 391\u2013431 (1985)","journal-title":"Commun. Partial Differ. Equ."},{"key":"1085_CR5","volume-title":"Finite Elements","author":"D Braess","year":"2011","unstructured":"Braess, D.: Finite Elements, 5th edn. Cambridge University Press, Cambridge (2011)","edition":"5"},{"issue":"02","key":"1085_CR6","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1051\/m2an\/2011038","volume":"46","author":"E Cances","year":"2012","unstructured":"Cances, E., Chakir, R., Maday, Y.: Numerical analysis of the planewave discretization of some orbital-free and Kohn\u2013Sham models. ESAIM Math. Model. Numer. Anal. 46(02), 341\u2013388 (2012)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"issue":"1\u20133","key":"1085_CR7","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1007\/s10915-010-9358-1","volume":"45","author":"E Cances","year":"2010","unstructured":"Cances, E., Chakir, R., Maday, Y.: Numerical analysis of nonlinear eigenvalue problems. J. Sci. Comput. 45(1\u20133), 90\u2013117 (2010)","journal-title":"J. Sci. Comput."},{"key":"1085_CR8","doi-asserted-by":"publisher","first-page":"1073","DOI":"10.1007\/s00211-016-0826-x","volume":"135","author":"C Canuto","year":"2017","unstructured":"Canuto, C., Nochetto, R.H., Stevenson, R.P., Verani, M.: Convergence and optimality of $${\\bf {hp}}$$- AFEM. Numer. Math. 135, 1073\u20131119 (2017)","journal-title":"Numer. Math."},{"key":"1085_CR9","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198503972.001.0001","volume-title":"Perfect Incompressible Fluids Oxford Lecture Series in Mathematics and its Applications","author":"JY Chemin","year":"1998","unstructured":"Chemin, J.Y.: Perfect Incompressible Fluids Oxford Lecture Series in Mathematics and its Applications, vol. 14. The Clarendon Press\/Oxford University Press, New York (1998)"},{"key":"1085_CR10","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s11401-018-1060-x","volume":"39","author":"J Chu","year":"2018","unstructured":"Chu, J., Coron, J.-M., Shang, P., Tang, S.-X.: Gevrey class regularity of a semigroup associated with a nonlinear Korteweg\u2013de Vries equation. Chin. Ann. Math. Ser. B 39, 201\u2013212 (2018)","journal-title":"Chin. Ann. Math. Ser. B"},{"key":"1085_CR11","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1051\/m2an\/2010061","volume":"45","author":"A Chernov","year":"2011","unstructured":"Chernov, A., von Petersdorff, T., Schwab, Ch.: Exponential convergence of $$hp$$ quadrature for integral operators with Gevrey kernels. ESAIM Math. Model. Numer. Anal. 45, 387\u2013422 (2011)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"1085_CR12","doi-asserted-by":"crossref","unstructured":"Costabel, M., Dauge, M., Nicaise, S.: Mellin analysis of weighted Sobolev spaces with nonhomogeneous norms on cones. In: Around the Research of V.G. Maz\u2019ya, pp. 105\u2013136. Springer, New York (2010)","DOI":"10.1007\/978-1-4419-1341-8_4"},{"issue":"8","key":"1085_CR13","doi-asserted-by":"publisher","first-page":"1250015","DOI":"10.1142\/S0218202512500157","volume":"22","author":"M Costabel","year":"2012","unstructured":"Costabel, M., Dauge, M., Nicaise, S.: Analytic regularity for linear elliptic systems in polygons and polyhedra. Math. Models Methods Appl. Sci. 22(8), 1250015 (2012)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"4","key":"1085_CR14","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1142\/S0218202505000480","volume":"15","author":"M Costabel","year":"2005","unstructured":"Costabel, M., Dauge, M., Schwab, C.: Exponential convergence of $$hp$$-FEM for Maxwell\u2019s equations with weighted regularization in polygonal domains. Math. Models Methods Appl. Sci. 15(4), 575\u2013622 (2005)","journal-title":"Math. Models Methods Appl. Sci."},{"key":"1085_CR15","doi-asserted-by":"crossref","unstructured":"Demkowicz, L., Kurtz, J.J., Pardo, D., Paszynski, M., Rachowicz, W., Zdunek, A.: Computing with hp-Adaptive Finite Elements. Vol. 2: Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications. Chapman and Hall\/CRC Applied Mathematics and Nonlinear Science Series. Boca Raton, (2007)","DOI":"10.1201\/9781420011692"},{"key":"1085_CR16","unstructured":"Frauenfelder, P.: hp-Finite Element Methods on Anisotropically, Locally Refined Meshes in Three Dimensions with Stochastic Data. PhD thesis, Swiss Federal Institute of Technology. Diss. ETH No. 15750, 2004. http:\/\/e-collection.library.ethz.ch\/. Accessed 31 Mar 2018"},{"issue":"31","key":"1085_CR17","doi-asserted-by":"publisher","first-page":"315208","DOI":"10.1088\/1751-8113\/42\/31\/315208","volume":"42","author":"S Fournais","year":"2009","unstructured":"Fournais, S., S\u00f8rensen, T.O., Hoffmannn-Ostenhof, M., Hoffmann-Ostenhof, T.: Analytic structure of solutions to multiconfiguration equations. J. Phys. A 42(31), 315208 (2009)","journal-title":"J. Phys. A"},{"issue":"6","key":"1085_CR18","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1007\/BF01389734","volume":"49","author":"W Gui","year":"1986","unstructured":"Gui, W., Babu\u0161ka, I.: The $$h,\\;p$$ and $$h$$-$$p$$ versions of the finite element method in $$1$$ dimension. II. The error analysis of the $$h$$- and $$h$$-$$p$$ versions. Numer. Math. 49(6), 613\u2013657 (1986)","journal-title":"Numer. Math."},{"issue":"21\u201341","key":"1085_CR19","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1007\/BF00272624","volume":"1","author":"BQ Guo","year":"1986","unstructured":"Guo, B.Q., Babu\u0161ka, I.: The $$hp$$-version of the finite element method. Part I: the basic approximation results, part II: general results and applications. Comput. Mech. 1(21\u201341), 203\u2013220 (1986)","journal-title":"Comput. Mech."},{"issue":"1","key":"1085_CR20","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1017\/S0308210500023520","volume":"127","author":"BQ Guo","year":"1997","unstructured":"Guo, B.Q., Babu\u0161ka, I.: Regularity of the solutions for elliptic problems on nonsmooth domains in $${R}^3$$. I. Countably normed spaces on polyhedral domains. Proc. R. Soc. Edinburgh Sect. A 127(1), 77\u2013126 (1997)","journal-title":"Proc. R. Soc. Edinburgh Sect. A"},{"key":"1085_CR21","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1016\/j.cam.2005.02.018","volume":"119","author":"BQ Guo","year":"2006","unstructured":"Guo, B.Q., Schwab, C.: Analytic regularity of Stokes flow on polygonal domains in countably weighted Sobolev spaces. J. Comp. Appl. Math. 119, 487\u2013519 (2006)","journal-title":"J. Comp. Appl. Math."},{"key":"1085_CR22","doi-asserted-by":"publisher","first-page":"767","DOI":"10.1007\/s00211-017-0918-2","volume":"138","author":"B Holm","year":"2018","unstructured":"Holm, B., Wihler, T.P.: Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up. Numer. Math. 138, 767\u2013799 (2018)","journal-title":"Numer. Math."},{"key":"1085_CR23","volume-title":"Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs","author":"VA Kozlov","year":"1997","unstructured":"Kozlov, V.A., Maz\u2019ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs, vol. 52. American Mathematical Society, Providence (1997)"},{"issue":"13","key":"1085_CR24","doi-asserted-by":"publisher","first-page":"4515","DOI":"10.1016\/j.jcp.2012.03.009","volume":"231","author":"L Lin","year":"2012","unstructured":"Lin, L., Lu, J., Ying, L., Weinan, E.: Optimized local basis set for Kohn-Sham density functional theory. J. Comput. Phys. 231(13), 4515\u20134529 (2012)","journal-title":"J. Comput. Phys."},{"key":"1085_CR25","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1017\/S0962492913000068","volume":"22","author":"M Luskin","year":"2013","unstructured":"Luskin, M., Ortner, C.: Atomistic-to-continuum coupling. Acta Numer. 22, 397\u2013508 (2013)","journal-title":"Acta Numer."},{"key":"1085_CR26","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/978-3-642-41401-5_14","volume-title":"Partial Differential Equations: Theory, Control and Approximation","author":"Y Maday","year":"2014","unstructured":"Maday, Y.: $$h-P$$ finite element approximation for full-potential electronic structure calculations. In: Ciarlet, P.G. (ed.) Partial Differential Equations: Theory, Control and Approximation, pp. 349\u2013377. Springer, Dordrecht (2014)"},{"key":"1085_CR27","unstructured":"Marcati, C.: Discontinuous $$hp$$ FEM for elliptic eigenvalue problems with singular potentials (with applications to quantum chemistry). Th\u00e8se de doctorat de l\u2019Universit\u00e9 P. et M. Curie, Paris (2018)"},{"key":"1085_CR28","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1137\/S0036142994267552","volume":"34","author":"R Munoz-Sola","year":"1997","unstructured":"Munoz-Sola, R.: Polynomial liftings on a tetrahedron and applications to the $$hp$$-FEM in three dimensions. SIAM J. Numer. Anal. 34, 282\u2013314 (1997)","journal-title":"SIAM J. Numer. Anal."},{"key":"1085_CR29","doi-asserted-by":"crossref","unstructured":"Samarskii, A.A., Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P., Blow-up in quasilinear parabolic equations de Gruyter Expositions in Mathematics, 19: Translated from the 1987 Russian Original by Michael Grinfeld and Revised by the Authors. Walter de Gruyter & Co., Berlin (1995)","DOI":"10.1515\/9783110889864"},{"issue":"4","key":"1085_CR30","doi-asserted-by":"publisher","first-page":"2005","DOI":"10.1137\/090774276","volume":"51","author":"D Sch\u00f6tzau","year":"2013","unstructured":"Sch\u00f6tzau, D., Schwab, C., Wihler, T.P.: $$hp$$-dGFEM for elliptic problems in polyhedra II: exponential convergence. SIAM J. Numer. Anal. 51(4), 2005\u20132035 (2013). (extended version in Technical Report 2009-29, Seminar for Applied Mathematics, ETH Z\u00fcrich)","journal-title":"SIAM J. Numer. Anal."},{"key":"1085_CR31","volume-title":"$$p$$ and $$hp$$-FEM, Theory and Applications in Solid and Fluid Mechanics","author":"Ch Schwab","year":"1998","unstructured":"Schwab, Ch.: $$p$$ and $$hp$$-FEM, Theory and Applications in Solid and Fluid Mechanics. Oxford University Press, Oxford (1998)"},{"key":"1085_CR32","doi-asserted-by":"crossref","unstructured":"Schwab, Ch., Exponential convergence of simplicial $$hp$$-FEM for $$H^1$$-functions with isotropic singularities. In: Proceedings of Conference Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014, Springer Lecture Notes CSE, vol. 106, pp. 435\u2013443. Springer (2015)","DOI":"10.1007\/978-3-319-19800-2_40"},{"issue":"190","key":"1085_CR33","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1090\/S0025-5718-1990-1011446-7","volume":"54","author":"LR Scott","year":"1990","unstructured":"Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483\u2013493 (1990)","journal-title":"Math. Comput."},{"key":"1085_CR34","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1090\/S0025-5718-07-01959-X","volume":"77","author":"R Stevenson","year":"2008","unstructured":"Stevenson, R.: The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77, 227\u2013241 (2008)","journal-title":"Math. Comput."},{"issue":"2","key":"1085_CR35","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1007\/BF02684475","volume":"59","author":"CT Traxler","year":"1997","unstructured":"Traxler, C.T.: An algorithm for adaptive mesh refinement in n dimensions. Computing 59(2), 115\u2013137 (1997)","journal-title":"Computing"},{"key":"1085_CR36","doi-asserted-by":"publisher","first-page":"2765","DOI":"10.1016\/S0045-7825(03)00294-9","volume":"192","author":"T Warburton","year":"2003","unstructured":"Warburton, T., Hesthaven, J.S.: On the constants in $$hp$$-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Eng. 192, 2765\u20132773 (2003)","journal-title":"Comput. Methods Appl. Mech. Eng."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-019-01085-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-019-01085-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-019-01085-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,27]],"date-time":"2024-07-27T04:05:44Z","timestamp":1722053144000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-019-01085-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,18]]},"references-count":36,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,2]]}},"alternative-id":["1085"],"URL":"https:\/\/doi.org\/10.1007\/s00211-019-01085-z","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,11,18]]},"assertion":[{"value":"7 August 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 August 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 November 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}