Exponential convergence of hp FEM for spectral fractional diffusion in polygons

Abstract

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 Hs(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∈ Hs(Ω) of Lsu= 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.