Exponential Convergence of hp FEM for Spectral Fractional Diffusion in Polygons
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For the spectral fractional diffusion operator of order 2s∈ (0,2) in bounded, curvilinear polygonal domains Ω 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 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 in Ω. 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^s u = 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 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 in Ω. The present analysis for either approach extends to polygonal subsets ℳ of analytic, compact 2-manifolds 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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