A finite element discretization of fractional problems using graded meshes

Abstract

In this paper, we deal with the finite element approximation of a homogeneous Dirichlet boundary value problem for fractional powers of symmetric second--order elliptic operators on a two--dimensional domain $Omega$. We employ the diagonalization technique introduced in [Banjai, Melenk, Nochetto, Otárola, Salgado, Schwab, Foundations of Computational Mathematics (2019) 19: 901--962], which proposes a semi--discretization in the extended variable of a truncated Caffarelli--Silvestre extension. This approach decouples the problem into the solution of independent second--order reaction--diffusion equations in $Omega$, several of which extcolor{blue}{may} become singularly perturbed. For the case where $Omega=(0,1)^2$, we propose to approximate all the decoupled problems by bilinear finite elements over a unique layer adapted, suitably graded, rectangular mesh, which can be designed independently of the eventual singular perturbation parameters. We prove the convergence of the proposed scheme and show numerical examples confirming the theoretical results.