Robust Multigrid on Adaptive Meshes for the Integral Fractional Laplacian

Abstract

We analyze a geometric multigrid method on locally refined mesh hierarchies for non-local variational problems, focusing on the integral fractional Laplacian. For newest-vertex-bisection-generated hierarchies, we prove uniform contraction of the multigrid iteration independently of the refinement level, the smallest local mesh size, and the polynomial degree in the finite element discretization. The proof is based on an abstract Hilbert-space framework, a strengthened Cauchy--Schwarz estimate obtained from a near-/far-field decomposition of the non-local bilinear form, and a multilevel stable decomposition independent of the discretization parameters. Numerical experiments underline the theoretical findings.