hp-FEM for fractional diffusion
We study the Dirichlet problem for the integral fractional Laplacianin a polygon Ω with analytic right-hand side. We show the solution to be in a weighted analyticity class that captures both the analyticity of the solution in Ω and the singular behavior near the boundary. Near the boundary the solution has an anisotropic behavior: near edges but away from the corners, the solution is smooth in tangential direction and higher order derivatives in normal direction are singular at edges. At the corners, also higher order tangential derivatives are singular. This behavior is captured in terms of weights that are products of powers of the distances from edges and corners.
The proof of the weighted analytic regularity assertions is based on the Caffarelli-Silvestre extension, which realizes the non-local fractional Laplacianas a Dirichlet-to-Neumann map of a (degenerate) second order ellipticboundary value problem. This latter problem is in turn amenable tothe techniques developed for the study of the regularity of solution sof second order elliptic PDEs.
We employ our analytic regularity to show exponential convergence of high order finite element methods (hp-FEM) on meshes that are geometrically refined towards both edges and corners. The geometric refinement towards edges results in anisotropic meshes away from corners. The use of such anisotropic elements is crucial for the exponential convergence result. These mesh design principles are the same one as for hp-FEM discretizations of the Dirichlet spectral fractional Laplacian in polygons, for which [Banjai, Melenk, & Schwab] recently showed exponential convergence.