Constructive approximation on graded meshes for the integral fractional Laplacian

We consider the homogeneous Dirichlet problem for the integral fractional Laplacian. We prove optimal Sobolev regularity estimates in Lipschitz domains satisfying an exterior ball condition. 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.