Finite element approximation of the Sobolev constant

Abstract

Denoting by S the sharp constant in the Sobolev inequality in $${{\rm W}_0^{1,2}(B)}$$, being B the unit ball in $${\mathbb{R}^3}$$, and denoting by S h its approximation in a suitable finite element space, we show that S h converges to S as $${h\searrow0}$$with a polynomial rate of convergence. We provide both an upper and a lower bound on the rate of convergence, and present some numerical results.