Local convergence of the FEM for the integral fractional Laplacian

We provide for first order discretizations of the integral fractional Laplacian sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm

On the robust exponential convergence of hp finite element methods for problems with boundary layers

The hp version of the finite element method for a one-dimensional, singularly perturbed elliptic-elliptic model problem with analytic input data is considered. It is shown that the use of piecewise polynomials of degree p on a mesh consisting of three suitably chosen elements leads to robust exponential convergence, i.e., the exponential rate of convergence depends only on the input data and is independent of the perturbation paramete