Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons

Abstract

We prove exponential convergence in the energy norm of $hp$ finite element discretizations for the integral fractional diffusion operator of order $2s\in (0,2)$ subject to homogeneous Dirichlet boundary conditions in bounded polygonal domains $\Omega \subset {\mathbb{R}}^2$. Key ingredient in the analysis are the weighted analytic regularity from our previous work and meshes that feature anisotropic geometric refinement towards $\partial \Omega$.

References in corpus (15)

• Ten equivalent definitions of the fractional Laplace operator
• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• A fractional Laplace equation: regularity of solutions and Finite Element approximations
• Numerical methods for nonlocal and fractional models
• Numerical Methods for Fractional Diffusion
• Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver
• Regularity theory and high order numerical methods for the (1d)-Fractional Laplacian
• Tensor FEM for spectral fractional diffusion
• $\mathcal{H}$-matrix approximability of inverses of discretizations of the fractional Laplacian
• Local convergence of the FEM for the integral fractional Laplacian
• Weighted analytic regularity for the integral fractional Laplacian in polygons
• Exponential Convergence of $hp$ FEM for Spectral Fractional Diffusion in Polygons
• Robust exponential convergence of hp-FEM in balanced norms for singularly perturbed reaction-diffusion problems: corner domains
• An analysis of a butterfly algorithm
• Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D

Cited by in corpus (2)

• Fractional Dissipative PDEs
• Caloric functions and boundary regularity for the fractional Laplacian in Lipschitz open sets