Local Energy Estimates for the Fractional Laplacian

The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It known that solutions to the Dirichlet problem involving such an operator exhibit algebraic boundary singularity regardless domain regularity. This, in turn, deteriorates global regularity and as result convergence rate numerical solutions. For finite element discretizations, we derive local error estimates $H^s$-seminorm show optimal rates interior by only assuming meshes be shape-regular. These quantify fact reduced approximation concentrated near domain. We illustrate our theoretical results with several examples.