For the spectral fractional diffusion operator of order 2s, s∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$s \in (0,1)$$\end{document}, in bounded, curvilinear polygonal domains Ω⊂R2\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega \subset {{\mathbb {R}}}^2$$\end{document} we prove exponential convergence of two classes of hp discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm Hs(Ω)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {H}}^s(\varOmega )$$\end{document}. The first hp discretization is based on writing the solution as a co-normal derivative of a 2+1\documentclass[12pt]{minimal}
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\begin{document}$$2+1$$\end{document}-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 inΩ\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega $$\end{document}. Leveraging results on robust exponential convergence of hp-FEM for second order, linear reaction diffusion boundary value problems in Ω\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega $$\end{document}, exponential convergence rates for solutions u∈Hs(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathbb {H}}^s(\varOmega )$$\end{document} of Lsu=f\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}^s u = f$$\end{document} follow. Key ingredient in this hp-FEM are boundary fitted meshes with geometric mesh refinement towards∂Ω\documentclass[12pt]{minimal}
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\begin{document}$$\partial \varOmega $$\end{document}. The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of L-s\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}^{-s}$$\end{document} combined with hp-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations inΩ\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega $$\end{document}. The present analysis for either approach extends to (polygonal subsets M~\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{\mathcal {M}}}$$\end{document} of) analytic, compact 2-manifolds M\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {M}}$$\end{document}, parametrized by a global, analytic chart χ\documentclass[12pt]{minimal}
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\begin{document}$$\chi $$\end{document} with polygonal Euclidean parameter domain Ω⊂R2\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega \subset {{\mathbb {R}}}^2$$\end{document}. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogorov n-widths of solution sets for spectral fractional diffusion in curvilinear polygons and for analytic source terms are deduced.
