Weighted analytic regularity for the integral fractional Laplacian in polygons and application to $hp$-FEM
A talk in the Nonlocal Equations: Analysis and Numerics series by
Markus Melenk from Wien
| Abstract: | We study the Dirichlet problem for the integral fractional Laplacian in a polygon $\Omega$ 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 $\Omega$ 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 are products of powers of the distance from edges and corners. The proof of the regularity assertions is based on the Caffarelli-Silvestre extension, which realizes the non-local fractional Laplacian as a Dirichlet-to-Neumann map of a (degenerate) elliptic boundary problem. Our weighted analytic regularity can be used to show exponential convergence of high order finite element methods ($hp$-FEM) on suitably refined meshes. The refinement is towards both the edges and the corners with the refinement towards edges being anisotropic away corners. This work is joint with Markus Faustmann (TU Wien), Carlo Marcati (ETHZ), and Christoph Schwab (ETHZ). Within the CRC this talk is associated to the project(s): A7 |