Constructive Approximation on Graded Meshes for the Integral Fractional Laplacian

We consider the homogeneous Dirichlet problem for the integral fractional Laplacian (-delta)(s). We prove optimal Sobolev regularity estimates in Lipschitz domains pro-vided the solution is C-s 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.