{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T17:38:40Z","timestamp":1772473120422,"version":"3.50.1"},"reference-count":60,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T00:00:00Z","timestamp":1669939200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T00:00:00Z","timestamp":1669939200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2023,1]]},"abstract":"Abstract<\/jats:title>For the spectral fractional diffusion operator of order 2s<\/jats:italic>, $$s \\in (0,1)$$<\/jats:tex-math>\n \n s<\/mml:mi>\n \u2208<\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, in bounded, curvilinear polygonal domains $$\\varOmega \\subset {{\\mathbb {R}}}^2$$<\/jats:tex-math>\n \n \u03a9<\/mml:mi>\n \u2282<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> we prove exponential convergence of two classes of hp<\/jats:italic> discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm $${\\mathbb {H}}^s(\\varOmega )$$<\/jats:tex-math>\n \n \n \n H<\/mml:mi>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u03a9<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The first hp<\/jats:italic> discretization is based on writing the solution as a co-normal derivative of a $$2+1$$<\/jats:tex-math>\n \n 2<\/mml:mn>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-dimensional local, linear elliptic boundary value problem, to which an hp<\/jats:italic>-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations in<\/jats:italic>$$\\varOmega $$<\/jats:tex-math>\n \u03a9<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Leveraging results on robust exponential convergence of hp<\/jats:italic>-FEM for second order, linear reaction diffusion boundary value problems in $$\\varOmega $$<\/jats:tex-math>\n \u03a9<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, exponential convergence rates for solutions $$u\\in {\\mathbb {H}}^s(\\varOmega )$$<\/jats:tex-math>\n \n u<\/mml:mi>\n \u2208<\/mml:mo>\n \n \n H<\/mml:mi>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u03a9<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of $$\\mathcal {L}^s u = f$$<\/jats:tex-math>\n \n \n \n L<\/mml:mi>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msup>\n u<\/mml:mi>\n =<\/mml:mo>\n f<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> follow. Key ingredient in this hp<\/jats:italic>-FEM are boundary fitted meshes with geometric mesh refinement towards<\/jats:italic>$$\\partial \\varOmega $$<\/jats:tex-math>\n \n \u2202<\/mml:mi>\n \u03a9<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of $$\\mathcal {L}^{-s}$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n <\/mml:mrow>\n \n -<\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> combined with hp<\/jats:italic>-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in<\/jats:italic>$$\\varOmega $$<\/jats:tex-math>\n \u03a9<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The present analysis for either approach extends to (polygonal subsets $${\\widetilde{\\mathcal {M}}}$$<\/jats:tex-math>\n \n M<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of) analytic, compact 2-manifolds $${\\mathcal {M}}$$<\/jats:tex-math>\n M<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, parametrized by a global, analytic chart $$\\chi $$<\/jats:tex-math>\n \u03c7<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with polygonal Euclidean parameter domain $$\\varOmega \\subset {{\\mathbb {R}}}^2$$<\/jats:tex-math>\n \n \u03a9<\/mml:mi>\n \u2282<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogorov n<\/jats:italic>-widths of solution sets for spectral fractional diffusion in curvilinear polygons and for analytic source terms are deduced.<\/jats:p>","DOI":"10.1007\/s00211-022-01329-5","type":"journal-article","created":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T16:33:10Z","timestamp":1669998790000},"page":"1-47","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Exponential convergence of hp FEM for spectral fractional diffusion in polygons"],"prefix":"10.1007","volume":"153","author":[{"given":"Lehel","family":"Banjai","sequence":"first","affiliation":[]},{"given":"Jens M.","family":"Melenk","sequence":"additional","affiliation":[]},{"given":"Christoph","family":"Schwab","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,12,2]]},"reference":[{"key":"1329_CR1","doi-asserted-by":"crossref","unstructured":"Ainsworth, M., Glusa, C.: Hybrid finite element-spectral method for the fractional Laplacian: approximation theory and efficient solver. SIAM J. Sci. Comput. 40(4), A2383\u2013A2405 (2018)","DOI":"10.1137\/17M1144696"},{"key":"1329_CR2","doi-asserted-by":"crossref","unstructured":"Antil, H., Chen, Y., Narayan, A.: Reduced basis methods for fractional Laplace equations via extension. SIAM J. Sci. Comput. 41(6), A3552\u2013A3575 (2019)","DOI":"10.1137\/18M1204802"},{"key":"1329_CR3","doi-asserted-by":"crossref","unstructured":"Apel, T., Melenk, J.M.: Interpolation and quasi-interpolation in $$h$$- and $$hp$$-version finite element spaces. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, 2nd edn., pp. 1\u201333. Wiley, Chichester, UK (2018). https:\/\/www.asc.tuwien.ac.at\/preprint\/2015\/asc39x2015.pdf","DOI":"10.1002\/9781119176817.ecm2002m"},{"key":"1329_CR4","doi-asserted-by":"crossref","unstructured":"Aubin, T.: Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics, Springer-Verlag, Berlin (1998)","DOI":"10.1007\/978-3-662-13006-3"},{"key":"1329_CR5","doi-asserted-by":"crossref","unstructured":"Babu\u0161ka, I., Guo, B.Q.: The $$h-p$$ version of the finite element method. Part 1: the basic approximation results. Comput Mech 1, 21\u201341 (1986)","DOI":"10.1007\/BF00298636"},{"key":"1329_CR6","doi-asserted-by":"crossref","unstructured":"Babu\u0161ka, I., Guo., B.Q.: The $$h-p$$ version of the finite element method. Part 2: general results and applications. Comput. Mech. 1, 203\u2013220 (1986)","DOI":"10.1007\/BF00272624"},{"key":"1329_CR7","doi-asserted-by":"crossref","unstructured":"Balakrishnan, A.V.: Fractional powers of closed operators and the semigroups generated by them. Pac. J. Math. 10, 419\u2013437 (1960)","DOI":"10.2140\/pjm.1960.10.419"},{"key":"1329_CR8","doi-asserted-by":"crossref","unstructured":"Banjai, L., Melenk, J.M., Schwab, C.: $$hp$$-FEM for reaction-diffusion equations. II: robust exponential convergence for multiple length scales in corner domains. Technical Report 2020-28, Seminar for Applied Mathematics, ETH Z\u00fcrich, Switzerland, 2020. (to appear in IMA Journal of Numerical Analysis 2022)","DOI":"10.1093\/imanum\/drac070"},{"key":"1329_CR9","doi-asserted-by":"crossref","unstructured":"Banjai, L., Melenk, J.M., Nochetto, R.H., Ot\u00e1rola, E., Salgado, A.J., Schwab, C.: Tensor FEM for spectral fractional diffusion. Found. Comput. Math. 19(4), 901\u2013962 (2019)","DOI":"10.1007\/s10208-018-9402-3"},{"key":"1329_CR10","doi-asserted-by":"crossref","unstructured":"Bonito, A., Lei, W., Pasciak, J.E.: On sinc quadrature approximations of fractional powers of regularly accretive operators. J. Num. Math. 27(2), 57\u201368 (2019)","DOI":"10.1515\/jnma-2017-0116"},{"key":"1329_CR11","doi-asserted-by":"crossref","unstructured":"Bonito, A., Pasciak, J.E.: Numerical approximation of fractional powers of elliptic operators. Math. Comp. 84(295), 2083\u20132110 (2015)","DOI":"10.1090\/S0025-5718-2015-02937-8"},{"key":"1329_CR12","doi-asserted-by":"crossref","unstructured":"Bonito, A., Borthagaray, J.P., Nochetto, R.H., Ot\u00e1rola, E., Salgado, A.J.: Numerical methods for fractional diffusion. Comput. Vis. Sci. 19(5\u20136), 19\u201346 (2018)","DOI":"10.1007\/s00791-018-0289-y"},{"key":"1329_CR13","doi-asserted-by":"crossref","unstructured":"Bonito, A., Guignard, D., Zhang, A.R.: Reduced basis approximations of the solutions to spectral fractional diffusion problems. J. Numer. Math. 28(3), 147\u2013160 (2020)","DOI":"10.1515\/jnma-2019-0053"},{"key":"1329_CR14","unstructured":"Bonito, A., Lei, W.: Approximation of the spectral fractional powers of the Laplace-Beltrami operator, (2021). arXiv:2101.05141"},{"key":"1329_CR15","doi-asserted-by":"crossref","unstructured":"Cabr\u00e9, X., Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224(5), 2052\u20132093 (2010)","DOI":"10.1016\/j.aim.2010.01.025"},{"key":"1329_CR16","doi-asserted-by":"crossref","unstructured":"Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Part. Diff. Eqs. 32(7\u20139), 1245\u20131260 (2007)","DOI":"10.1080\/03605300600987306"},{"key":"1329_CR17","doi-asserted-by":"crossref","unstructured":"Caffarelli, L.A., Stinga, P.R.: Fractional elliptic equations, Caccioppoli estimates and regularity. Ann. Inst.H. Poincar\u00e9 Anal. Non Lin\u00e9aire 33(3), 767\u2013807 (2016)","DOI":"10.1016\/j.anihpc.2015.01.004"},{"key":"1329_CR18","doi-asserted-by":"crossref","unstructured":"Cusimano, N., del Teso, F., Gerardo-Giorda, L., Pagnini, G.: Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions. SIAM J. Numer. Anal. 56(3), 1243\u20131272 (2018)","DOI":"10.1137\/17M1128010"},{"key":"1329_CR19","doi-asserted-by":"crossref","unstructured":"Danczul, T., Hofreither, C.: On rational Krylov and reduced basis methods for fractional diffusion. J. Numer. Math. 30(2), 121\u2013140 (2022)","DOI":"10.1515\/jnma-2021-0032"},{"key":"1329_CR20","doi-asserted-by":"crossref","unstructured":"Danczul, T., Sch\u00f6berl, J.: A reduced basis method for fractional diffusion operators II. J. Numer. Math. 29(4), 269\u2013287 (2021)","DOI":"10.1515\/jnma-2020-0042"},{"key":"1329_CR21","doi-asserted-by":"crossref","unstructured":"Danczul, T., Sch\u00f6berl, J.: A reduced basis method for fractional diffusion operators I. Numer. Math. 151(2), 369\u2013404 (2022)","DOI":"10.1007\/s00211-022-01287-y"},{"key":"1329_CR22","doi-asserted-by":"crossref","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, volume 1939 of Lectures Notes in Mathematics. Springer Verlag, (2008)","DOI":"10.1007\/978-3-540-78319-0_3"},{"key":"1329_CR23","doi-asserted-by":"crossref","unstructured":"Demkowicz, L., Kurtz, J., Pardo, D., Paszy\u0144ski, M., Rachowicz, W., Zdunek, A.: Computing with $$hp$$-adaptive finite elements. Vol. 2. Chapman & Hall\/CRC Applied Mathematics and Nonlinear Science Series. Chapman & Hall\/CRC, Boca Raton, FL, (2008). Frontiers: three dimensional elliptic and Maxwell problems with applications","DOI":"10.1201\/9781420011692"},{"key":"1329_CR24","doi-asserted-by":"crossref","unstructured":"Faustmann, M., Melenk, J.M.: Robust exponential convergence of $$hp$$-FEM in balanced norms for singularly perturbed reaction-diffusion problems: corner domains. Comput. Math. Appl. 74(7), 1576\u20131589 (2017)","DOI":"10.1016\/j.camwa.2017.03.015"},{"key":"1329_CR25","doi-asserted-by":"crossref","unstructured":"Gavrilyuk, I.P., Hackbusch, W., Khoromskij, B.N.: Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems. Computing 74(2), 131\u2013157 (2005)","DOI":"10.1007\/s00607-004-0086-y"},{"issue":"5","key":"1329_CR26","doi-asserted-by":"publisher","first-page":"2505","DOI":"10.1137\/070700607","volume":"46","author":"N Hale","year":"2008","unstructured":"Hale, N., Higham, N.J., Trefethen, L.N.: Computing $${ A}^\\alpha, \\log ({ A})$$, and related matrix functions by contour integrals. SIAM J. Numer. Anal. 46(5), 2505\u20132523 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"1329_CR27","doi-asserted-by":"crossref","unstructured":"Harizanov, S., Lazarov, R., Margenov, S., Marinov, P., Vutov, Y.: Optimal solvers for linear systems with fractional powers of sparse SPD matrices. Numer. Linear Algebra Appl. 25(5), e2167, 24 (2018)","DOI":"10.1002\/nla.2167"},{"key":"1329_CR28","doi-asserted-by":"crossref","unstructured":"Harizanov, S., Lazarov, R., Margenov, S., Marinov, P., Pasciak, J.: Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation. J. Comput. Phys. 408, 109285, 21 (2020)","DOI":"10.1016\/j.jcp.2020.109285"},{"key":"1329_CR29","doi-asserted-by":"crossref","unstructured":"Helin, T., Lassas, M., Ylinen, L., Zhang, Z.: Inverse problems for heat equation and space-time fractional diffusion equation with one measurement. J. Differ. Equ. 269(9), 7498\u20137528 (2020)","DOI":"10.1016\/j.jde.2020.05.022"},{"key":"1329_CR30","doi-asserted-by":"crossref","unstructured":"Hofreither, C.: A unified view of some numerical methods for fractional diffusion. Comput. Math. Appl. 80(2), 332\u2013350 (2020)","DOI":"10.1016\/j.camwa.2019.07.025"},{"key":"1329_CR31","doi-asserted-by":"crossref","unstructured":"Karkulik, M., Melenk, J.M.: $$\\cal{H} $$-matrix approximability of inverses of discretizations of the fractional Laplacian. Adv. Comput. Math. 45(5\u20136), 2893\u20132919 (2019)","DOI":"10.1007\/s10444-019-09718-5"},{"key":"1329_CR32","unstructured":"Kre\u012dn, S.\u00a0G.: Interpolation of linear operators, and properties of the solutions of elliptic equations. In: Elliptische Differentialgleichungen, Band II, pages 155\u2013166. Schriftenreihe Inst. Math. Deutsch. Akad. Wissensch. Berlin, Reihe A, Heft 8. Akademie-Verlag, Berlin, (1971)"},{"key":"1329_CR33","doi-asserted-by":"crossref","unstructured":"Lischke, A., Pang, G., Gulian, M., Song, F., Glusa, C., Zheng, X., Mao, Z., Cai, W., Meerschaert, M.M., Ainsworth, M., Karniadakis, G.E.: What is the fractional Laplacian? A comparative review with new results. J. Comput. Phys. 404, 109009, 62 (2020)","DOI":"10.1016\/j.jcp.2019.109009"},{"key":"1329_CR34","doi-asserted-by":"crossref","unstructured":"Lynch, R.E., Rice, J.R., Thomas, D.H.: Direct solution of partial difference equations by tensor product methods. Numer. Math. 6, 185\u2013199 (1964)","DOI":"10.1007\/BF01386067"},{"key":"1329_CR35","unstructured":"McLean, W.: Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000)"},{"key":"1329_CR36","doi-asserted-by":"crossref","unstructured":"Meidner, D., Pfefferer, J., Sch\u00fcrholz, K., Vexler, B.: $$hp$$-finite elements for fractional diffusion. SIAM J. Numer. Anal. 56(4), 2345\u20132374 (2018)","DOI":"10.1137\/17M1135517"},{"key":"1329_CR37","doi-asserted-by":"crossref","unstructured":"Melenk, J., Schwab, C.: An hp finite element method for convection-diffusion problems in one dimension. IMA J. Numer. Anal. 19(3), 425\u2013453 (1999)","DOI":"10.1093\/imanum\/19.3.425"},{"key":"1329_CR38","doi-asserted-by":"crossref","unstructured":"Melenk, J.M., Rieder, A.: $$hp$$-FEM for the fractional heat equation. IMA J. Numer. Anal. 41(1), 412\u2013454 (2021)","DOI":"10.1093\/imanum\/drz054"},{"key":"1329_CR39","doi-asserted-by":"crossref","unstructured":"Melenk, J.M.: On the robust exponential convergence of $$hp$$ finite element method for problems with boundary layers. IMA J. Numer. Anal. 17(4), 577\u2013601 (1997)","DOI":"10.1093\/imanum\/17.4.577"},{"key":"1329_CR40","doi-asserted-by":"crossref","unstructured":"Melenk, J.M.: On $$n$$-widths for elliptic problems. J. Math. Anal. Appl. 247(1), 272\u2013289 (2000)","DOI":"10.1006\/jmaa.2000.6862"},{"key":"1329_CR41","doi-asserted-by":"crossref","unstructured":"Melenk, J.M.: $$hp$$-finite element methods for singular perturbations. Lecture Notes in Mathematics, vol. 1796. Springer-Verlag, Berlin (2002)","DOI":"10.1007\/b84212"},{"key":"1329_CR42","doi-asserted-by":"crossref","unstructured":"Melenk, J.M., Schwab, Ch.: $$hp$$ FEM for reaction-diffusion equations. I. Robust exponential convergence. SIAM J. Numer. Anal. 35(4), 1520\u20131557 (1998)","DOI":"10.1137\/S0036142997317602"},{"key":"1329_CR43","doi-asserted-by":"crossref","unstructured":"Melenk, J.M., Schwab, Ch.: Analytic regularity for a singularly perturbed problem. SIAM J. Math. Anal. 30(2), 379\u2013400 (1999)","DOI":"10.1137\/S0036141097317542"},{"key":"1329_CR44","doi-asserted-by":"crossref","unstructured":"Melenk, J.M., Xenophontos, C.: Robust exponential convergence of $$hp$$-FEM in balanced norms for singularly perturbed reaction-diffusion equations. Calcolo 53(1), 105\u2013132 (2016)","DOI":"10.1007\/s10092-015-0139-y"},{"key":"1329_CR45","doi-asserted-by":"crossref","unstructured":"Mori, M., Sugihara, M.: The double-exponential transformation in numerical analysis. J. Comput. Appl. Math. 127(1\u20132), 287\u2013296 (2001). (Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials)","DOI":"10.1016\/S0377-0427(00)00501-X"},{"key":"1329_CR46","doi-asserted-by":"crossref","unstructured":"Nochetto, R.H., Ot\u00e1rola, E., Salgado, A.J.: A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15(3), 733\u2013791 (2015)","DOI":"10.1007\/s10208-014-9208-x"},{"key":"1329_CR47","doi-asserted-by":"crossref","unstructured":"Quarteroni, A., Manzoni, A., Negri, F.: Reduced basis methods for partial differential equations, volume\u00a092 of Unitext. Springer, Cham, An introduction, La Matematica per il 3+2 (2016)","DOI":"10.1007\/978-3-319-15431-2"},{"key":"1329_CR48","unstructured":"Rieder, A.: Double exponential quadrature for fractional diffusion (2020). arXiv:2012.05588"},{"key":"1329_CR49","unstructured":"Roos, H.G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations, volume\u00a024 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, (2008). Convection-diffusion-reaction and flow problems"},{"key":"1329_CR50","doi-asserted-by":"crossref","unstructured":"Ros-Oton, X.: Nonlocal elliptic equations in bounded domains: a survey. Publ. Mat. 60(1), 3\u201326 (2016)","DOI":"10.5565\/PUBLMAT_60116_01"},{"key":"1329_CR51","doi-asserted-by":"crossref","unstructured":"Sch\u00f6berl, J.: NETGEN an advancing front 2D\/3D-mesh generator based on abstract rules. J. Comput. Visual. Sci. 1, 41\u201352 (1997)","DOI":"10.1007\/s007910050004"},{"key":"1329_CR52","unstructured":"Schwab, C.: $$p$$- and $$hp$$-Finite Element Methods. Numerical Mathematics and Scientific Computation. The Clarendon Press, Oxford University Press, New York, 1998. Theory and applications in solid and fluid mechanics"},{"key":"1329_CR53","doi-asserted-by":"crossref","unstructured":"Schwab, C., Suri, M.: The $$p$$ and $$hp$$ versions of the finite element method for problems with boundary layers. Math. Comp. 65(216), 1403\u20131429 (1996)","DOI":"10.1090\/S0025-5718-96-00781-8"},{"key":"1329_CR54","doi-asserted-by":"crossref","unstructured":"Schwab, C., Suri, M., Xenophontos, C.A.: The $$hp$$ finite element method for problems in mechanics with boundary layers. Comp. Meth. Appl. Mech. Eng. 157(3\u20134), 311\u2013333 (1998)","DOI":"10.1016\/S0045-7825(97)00243-0"},{"key":"1329_CR55","doi-asserted-by":"crossref","unstructured":"Song, F., Chuanju, X., Karniadakis, G.E.: Computing fractional Laplacians on complex-geometry domains: algorithms and simulations. SIAM J. Sci. Comput. 39(4), A1320\u2013A1344 (2017)","DOI":"10.1137\/16M1078197"},{"key":"1329_CR56","doi-asserted-by":"crossref","unstructured":"Stenger, F.: Numerical methods based on sinc and analytic functions. Springer Series in Computational Mathematics, vol. 20. Springer-Verlag, New York (1993)","DOI":"10.1007\/978-1-4612-2706-9"},{"key":"1329_CR57","doi-asserted-by":"crossref","unstructured":"Stinga, P.R., Torrea, J.L.: Extension problem and Harnack\u2019s inequality for some fractional operators. Comm. Partial Differ. Equ. 35(11), 2092\u20132122 (2010)","DOI":"10.1080\/03605301003735680"},{"key":"1329_CR58","unstructured":"Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. Johann Ambrosius Barth, Heidelberg (1995)"},{"key":"1329_CR59","doi-asserted-by":"crossref","unstructured":"Vabishchevich, P.N.: Numerically solving an equation for fractional powers of elliptic operators. J. Comput. Phys. 282, 289\u2013302 (2015)","DOI":"10.1016\/j.jcp.2014.11.022"},{"key":"1329_CR60","doi-asserted-by":"crossref","unstructured":"Vabishchevich, P.N.: Numerical solving unsteady space-fractional problems with the square root of an elliptic operator. Math. Model. Anal. 21(2), 220\u2013238 (2016)","DOI":"10.3846\/13926292.2016.1147000"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-022-01329-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-022-01329-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-022-01329-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T14:05:37Z","timestamp":1673273137000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-022-01329-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,2]]},"references-count":60,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,1]]}},"alternative-id":["1329"],"URL":"https:\/\/doi.org\/10.1007\/s00211-022-01329-5","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,2]]},"assertion":[{"value":"4 June 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 September 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 November 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 December 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}