Exponential convergence of hp FEM for spectral fractional diffusion in polygons

For the spectral fractional diffusion operator of order 2s, s∈ (0 , 1) , in bounded, curvilinear polygonal domains Ω⊂ R² 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 H^s(Ω). 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 inΩ. Leveraging results on robust exponential convergence of hp-FEM for second order, linear reaction diffusion boundary value problems in Ω, exponential convergence rates for solutions u∈ H^s(Ω) of L^su= f follow. Key ingredient in this hp-FEM are boundary fitted meshes with geometric mesh refinement towards∂Ω. The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of L^-s combined with hp-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations inΩ. The present analysis for either approach extends to (polygonal subsets M~ of) analytic, compact 2-manifolds M, parametrized by a global, analytic chart χ with polygonal Euclidean parameter domain Ω⊂ R². 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.