Exponential convergence of hp FEM for spectral fractional diffusion in polygons

Abstract

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