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dc.contributor.authorCusimano N.en_US
dc.contributor.authorDel Teso F.en_US
dc.contributor.authorGerardo-Giorda L.en_US
dc.description.abstractWe 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.sponsorshipSpanish project PGC2018-094522-B-I00; 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-2016-30148en_US
dc.publisherESAIM: Mathematical Modelling and Numerical Analysisen_US
dc.subjectFractional Laplacianen_US
dc.subjectBounded Domainen_US
dc.subjectBoundary Value Problemen_US
dc.subjectHomogeneous and Nonhomogeneous Boundary Conditionsen_US
dc.subjectHeat Semigroupen_US
dc.subjectIntegral Quadratureen_US
dc.titleNumerical approximations for fractional elliptic equations via the method of semigroupsen_US

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