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

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 $\tilde{\mathcal{M}}$ 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.

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Cited by in corpus (2)

• A fractional PDE model for turbulent velocity fields near solid walls
• Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons