Constructive approximation on graded meshes for the integral fractional Laplacian.

Abstract

We consider the homogeneous Dirichlet problem for the integral fractional Laplacian (−Δ)s. We prove optimal Sobolev regularity estimates in Lipschitz domains provided the solution is Cs up to the boundary. We present the construction of graded bisection meshes by a greedy algorithm and derive quasi-optimal convergence rates for approximations to the solution of such a problem by continuous piecewise linear functions. The nonlinear Sobolev scale dictates the relation between regularity and approximability.