Numerical approximations for fractional elliptic equationsviathe method of semigroups

Abstract

We provide a novel approach to the numerical solution of the family of nonlocal elliptic equations (−Δ)su=fin Ω, subject to some homogeneous boundary conditionsBon ∂Ω, wheres∈ (0,1), Ω ⊂ ℝnis a bounded domain, and (-Δ)sis the spectral fractional Laplacian associated toBon ∂Ω. We use the solution representation (−Δ)−sftogether 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 datumfin a suitable fractional Sobolev space of orderr≥ 0 and the discretization parameterh> 0, our numerical scheme converges asO(hr+2s), providing super quadratic convergence rates up toO(h4) for sufficiently regular data, or simplyO(h2s) for merelyf∈L2(Ω). We also extend the proposed framework to the case of nonhomogeneous boundary conditions and support our results with some illustrative numerical tests.