Finite element approximation of the Sobolev constant

Denoting by S the sharp constant in the Sobolev inequality in W₀^1,2(B){{rm W}_0^{1,2}(B)}, being B the unit ball in mathbbR³{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 hsearrow0{hsearrow0} 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.