dc.contributor.author Cusimano, N. dc.contributor.author Del Teso, F. dc.contributor.author Gerardo-Giorda, L. dc.date.accessioned 2020-02-13T09:11:31Z dc.date.available 2020-02-13T09:11:31Z dc.date.issued 2020 dc.identifier.uri http://hdl.handle.net/20.500.11824/1076 dc.description.abstract We provide a novel approach to the numerical solution of the family of nonlocal elliptic equations $(-\Delta)^su=f$ in $\Omega$, subject to some homogeneous boundary conditions $\mathcal{B}(u)=0$ on $\partial \Omega$, where $s\in(0,1)$, $\Omega\subset \mathbb{R}^n$ is a bounded domain, and $(-\Delta)^s$ is the spectral fractional Laplacian associated to $\mathcal{B}$ on $\partial \Omega$. We use the solution representation $(-\Delta)^{-s}f$ together with its singular integral expression given by the method of semigroups. By combining finite element discretizations for the heat semigroup with monotone quadratures for the singular integral we obtain accurate numerical solutions. Roughly speaking, given a datum $f$ in a suitable fractional Sobolev space of order $r\geq 0$ and the discretization parameter $h>0$, our numerical scheme converges as $O(h^{r+2s})$, providing super quadratic convergence rates up to $O(h^4)$ for sufficiently regular data, or simply $O(h^{2s})$ for merely $f\in L^2(\Omega)$. We also extend the proposed framework to the case of nonhomogeneous boundary conditions and support our results with some illustrative numerical tests. en_US dc.description.sponsorship Spanish project PGC2018-094522-B-I00; en_US Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway; MEC-Juan de la Cierva postdoctoral fellowship no. FJCI-2016-30148 dc.format application/pdf en_US dc.language.iso eng en_US dc.rights Reconocimiento-NoComercial-CompartirIgual 3.0 España en_US dc.rights.uri http://creativecommons.org/licenses/by-nc-sa/3.0/es/ en_US dc.subject Fractional Laplacian en_US dc.subject Bounded Domain en_US dc.subject Boundary Value Problem en_US dc.subject Homogeneous and Nonhomogeneous Boundary Conditions en_US dc.subject Heat Semigroup en_US dc.subject Integral Quadrature en_US dc.title Numerical approximations for fractional elliptic equations via the method of semigroups en_US dc.type info:eu-repo/semantics/article en_US dc.identifier.doi 10.1051/m2an/2019076 dc.relation.projectID EUS/BERC/BERC.2018-2021 en_US dc.rights.accessRights info:eu-repo/semantics/openAccess en_US dc.type.hasVersion info:eu-repo/semantics/acceptedVersion en_US dc.journal.title ESAIM: Mathematical Modelling and Numerical Analysis en_US
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