Galerkin Approximation of the Fractional Sobolev Constant

TL;DR

This paper derives precise estimates for the discrete fractional Sobolev constant using Galerkin approximation with linear elements in the unit ball, focusing on convergence rates and mesh regularity.

Contribution

It provides sharp bounds and convergence analysis for the fractional Sobolev inequality's discrete optimal constant via Galerkin methods in a specific geometric setting.

Findings

Established sharp estimates for the fractional Sobolev constant.

Analyzed convergence rates for Galerkin approximation with linear elements.

Utilized quasi-uniform, regular meshes in the unit ball.

Abstract

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