2019/10/26 by Nicole Cusimano, Félix del Teso, Luca Gerardo-Giorda +1 · 1 citation
Mathematics · #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #Fractional Differential Equations Solutions
paper · doi:10.1051/m2an/2019076
openalex publication_date 2019/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25
We provide a novel approach to the numerical solution of the family of nonlocal elliptic equations (−Δ) s u = f in Ω, subject to some homogeneous boundary conditions B on ∂Ω, where s ∈ (0,1), Ω ⊂ ℝ n 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 ( 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 ∈ L 2 (Ω). We also extend the proposed framework to the case of nonhomogeneous boundary conditions and support our results with some illustrative numerical tests.