Volume 54, Number 3, May-June 2020
|Page(s)||751 - 774|
|Published online||02 April 2020|
Numerical approximations for fractional elliptic equations via the method of semigroups
Basque Center for Applied Mathematics, Bilbao, Spain
2 Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
* Corresponding author: email@example.com
Accepted: 23 October 2019
We provide a novel approach to the numerical solution of the family of nonlocal elliptic equations (−Δ)su = f in Ω, subject to some homogeneous boundary conditions B on ∂Ω, where s ∈ (0,1), Ω ⊂ ℝn is a bounded domain, and (-Δ)s is the spectral fractional Laplacian associated to B on ∂Ω. We use the solution representation (−Δ)−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 ≥ 0 and the discretization parameter h > 0, our numerical scheme converges as O(hr+2s), providing super quadratic convergence rates up to O(h4) for sufficiently regular data, or simply O(h2s) for merely f ∈ L2 (Ω). We also extend the proposed framework to the case of nonhomogeneous boundary conditions and support our results with some illustrative numerical tests.
Mathematics Subject Classification: 35S15 / 65R20 / 65N15 / 65N25 / 41A55 / 35R11 / 26A33
Key words: Fractional Laplacian / bounded domain / boundary value problem / homogeneous and nonhomogeneous boundary conditions / heat semigroup / finite elements / integral quadrature
© EDP Sciences, SMAI 2020
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