  In this paper, we are concerned with the convergence rate of a FEM based
numerical scheme approximating extremal functions of the Sobolev inequality. We
prove that when the domain is polygonal and convex in $\R^2$, the convergence
of a finite element solution to an exact extremal function in $L^2$ and $H^1$
norms has the rates $O(h^2)$ and $O(h)$ respectively, where $h$ denotes the
mesh size of a triangulation of the domain.
