Exponential Convergence of $hp$ FEM for Spectral Fractional Diffusion in Polygons
Lehel Banjai, Jens M. Melenk, Christoph Schwab
TL;DR
The paper develops two robust $hp$-FEM frameworks for the spectral fractional diffusion operator ${\mathcal L}^s$ in curvilinear polygonal domains: (i) an extended formulation via the Caffarelli-Silvestre extension with a diagonalization that reduces to decoupled reaction-diffusion problems in the domain, and (ii) a Balakrishnan integral representation discretized by exponentially convergent sinc quadrature, also leading to decoupled local problems. By leveraging analytic regularity in the extended variable and robust $hp$-FEM error theory for singularly perturbed reaction-diffusion problems, both approaches achieve exponential convergence in the number of degrees of freedom, without requiring boundary compatibility. A key feature is the design of boundary-fitted, geometrically refined meshes toward edges and corners to resolve singularities and boundary layers, enabling robust performance across polygonal domains and even extending to analytic manifolds. The results include two discretization cases (Case A and Case B) with explicit error representations and complexity estimates, as well as extensions to fractional surface diffusion on manifolds and exponential bounds on Kolmogorov $n$-widths. Numerical experiments confirm the exponential rates and demonstrate practical efficiency over standard approaches, highlighting the potential for fast, high-accuracy simulations of fractional diffusion in complex geometries.
Abstract
For the spectral fractional diffusion operator of order $2s\in (0,2)$ 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 $\mathbb{H}^s(Ω)$. 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 $u\in \mathbb{H}^s(Ω)$ of $\mathcal{L}^s u = f$ follow. Key ingredient in this $hp$-FEM are boundary fitted meshes with geometric mesh refinement towards $\partialΩ$. The second discretization is based on exponentially convergent sinc quadrature approximations of the Balakrishnan integral representation of $\mathcal{L}^{-s}$, 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 $\widetilde{\mathcal{M}}$ of analytic, compact $2$-manifolds $\mathcal{M}$. 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.