numerical analysis

Galerkin Approximation of the Fractional Hardy Constant

arXiv:2607.26830

summary

The paper derives sharp estimates for the discrete optimal constant in the fractional Hardy inequality and analyzes convergence rates of a Galerkin finite element method with piecewise linear elements on quasi‑uniform meshes in smooth convex domains.

Abstract

We establish sharp estimates for the discrete optimal constant of the fractional Hardy Inequality in dimension $N \geq 1$ , with fractional exponent $s \in (0, min {1, \frac{N}{2}})$ . The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the computations are carried out in a bounded, convex and smooth domain containing the origin, for which we employ a quasi-uniform and regular mesh.

Topics & keywords

#fractional Hardy inequality #Galerkin approximation #finite element method #convergence rates #piecewise linear elements #mesh regularityfractional Hardy constantGalerkin methodpiecewise linear elementsquasi-uniform meshdiscrete optimal constantconvergence analysis