Local Convergence of the FEM for the Integral Fractional Laplacian

Abstract

We prove exponential convergence in the energy norm of hp finite element discretizations for the integral fractional diffusion operator of order 2s ∈ (0, 2) subject to homogeneous Dirichlet boundary conditions in bounded polygonal domains Ω ⊂ R 2 . Key ingredient in the analysis are the weighted analytic regularity from [15] and meshes that feature anisotropic geometric refinement towards ∂Ω.

Weighted Analytic Regularity for the Integral Fractional Laplacian in Polygons

“…[BLN22] and the references there), sharp regularity for variational solutions of (1.1) will imply corresponding convergence rate estimates of Galerkin approximations. Similar to the two-dimensional case, where analytic regularity of solutions to (1.1) on bounded, polygonal domains Ω, obtained in [FMMS22], implied exponential convergence bounds for corresponding hp FE Galerkin approximations in [FMMS23], the weighted analytic regularity estimates obtained in the present paper form the foundation for proving exponential rates of convergence of suitable families of hp-FEM in polyhedral domains Ω in a forthcoming work.…”

“…Directions for natural extensions of the present results in three space dimensions suggest themselves: first, the presently developed proof and the geometric structure of the weights in Ω should facilitate analogous weighted analytic regularity results for integral fractional diffusion such as (−∇ • A(x)∇) s , with an anisotropic diffusion coefficient A(•) being a uniformly positive definite d×d matrix, again with analytic in Ω entries. Likewise, the exponential convergence rate bound established in [FMMS23] in the two-dimensional setting will generalize to the presently considered, polyhedral setting, albeit with rate given by C exp(−bN 1/6 ), with N denoting the number of the degrees of freedom of the hp-FE subspace, and with constants b, C > 0 depending on Ω, f but not on N . Here, the larger number of geometric situations for ≥ 3 edges meeting in one, common vertex of ∂Ω will mandate significant extensions and additional technical issues as compared to the proof in [FMMS23].…”

“…Likewise, the exponential convergence rate bound established in [FMMS23] in the two-dimensional setting will generalize to the presently considered, polyhedral setting, albeit with rate given by C exp(−bN 1/6 ), with N denoting the number of the degrees of freedom of the hp-FE subspace, and with constants b, C > 0 depending on Ω, f but not on N . Here, the larger number of geometric situations for ≥ 3 edges meeting in one, common vertex of ∂Ω will mandate significant extensions and additional technical issues as compared to the proof in [FMMS23]. Details will be developed elsewhere.…”

Fractional Diffusion in the full space: decay and regularity

“…Numerical methods for fractional PDEs on bounded domains are fairly developed, as can e.g. be seen in the survey articles [BBN + 18, DDG + 20, LPG + 20] and we especially mention approximations based on the finite element method (FEM), [AB17, BMN + 19, ABH19,FKM22]. A key limitation to the FEM is the restriction to bounded computational domains.…”

Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D

“…In this article, we leverage our weighted analytic regularity estimates to design an exponentially convergent method by means of hp-finite element approximation in one dimension. The generalization to two dimensions is the topic of our follow-up work [12].…”