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For the spectral fractional diffusion operator of order $2s\in (0,2)$ inbounded, curvilinear polygonal domains $\Omega$ we prove exponentialconvergence of two classes of $hp$ discretizations under the assumption ofanalytic data, without any boundary compatibility, in the natural fractionalSobolev norm $\mathbb{H}^s(\Omega)$. The first $hp$ discretization is based onwriting the solution as a co-normal derivative of a $2+1$-dimensional local,linear elliptic boundary value problem, to which an $hp$-FE discretization isapplied. A diagonalization in the extended variable reduces the numericalapproximation of the inverse of the spectral fractional diffusion operator tothe numerical approximation of a system of local, decoupled, second orderreaction-diffusion equations in $\Omega$. Leveraging results on robustexponential convergence of $hp$-FEM for second order, linear reaction diffusionboundary value problems in $\Omega$, exponential convergence rates forsolutions $u\in \mathbb{H}^s(\Omega)$ of $\mathcal{L}^s u = f$ follow. Keyingredient in this $hp$-FEM are boundary fitted meshes with geometric meshrefinement towards $\partial\Omega$.  The second discretization is based on exponentially convergent sincquadrature approximations of the Balakrishnan integral representation of$\mathcal{L}^{-s}$, combined with $hp$-FE discretizations of a decoupled systemof 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 andwith incompatible data confirm the theoretical results.  Exponentially small bounds on Kolmogoroff $n$-widths of solutions sets forspectral fractional diffusion in polygons are deduced.

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