Exponential Convergence of $hp$ FEM for Spectral Fractional Diffusion in Polygons

TL;DR

This paper proves exponential convergence of $hp$ finite element methods for spectral fractional diffusion problems in polygons, using boundary fitted meshes and sinc quadrature, with confirmed numerical results.

Contribution

It introduces two novel $hp$ discretization approaches for spectral fractional diffusion in polygons, achieving exponential convergence without boundary compatibility conditions.

Findings

Exponential convergence rates are established for both discretization methods.

Numerical experiments confirm theoretical convergence in nonconvex polygons.

Exponential bounds on Kolmogoroff $n$-widths are derived.

Abstract

For the spectral fractional diffusion operator of order $2s\in (0,2)$ in bounded, curvilinear polygonal domains $\Omega$ we prove exponential convergence of two classes of $hp$ discretizations under the assumption of analytic data, without any boundary compatibility, in the natural fractional Sobolev norm $\mathbb{H}^s(\Omega)$. The first $hp$ discretization is based on writing the solution as a co-normal derivative of a $2+1$-dimensional local, linear elliptic boundary value problem, to which an $hp$-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations in $\Omega$. Leveraging results on robust exponential convergence of $hp$-FEM for second order, linear reaction diffusion boundary value problems in $\Omega$, exponential convergence rates for solutions $u\in \mathbb{H}^s(\Omega)$ of $\mathcal{L}^s u = f$ follow. Key ingredient in this $hp$-FEM are boundary fitted meshes with geometric mesh refinement towards $\partial\Omega$. The second discretization is based on exponentially convergent sinc quadrature approximations of the Balakrishnan integral representation of $\mathcal{L}^{-s}$, combined with $hp$-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in $\Omega$. The present analysis for either approach extends to polygonal subsets $\widetilde{\mathcal{M}}$ of analytic, compact $2$-manifolds $\mathcal{M}$. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogoroff $n$-widths of solutions sets for spectral fractional diffusion in polygons are deduced.