Optimal convergence rates for the finite element approximation of the Sobolev constant
Liviu I. Ignat, Enrique Zuazua
TL;DR
This work analyzes the $P1$ finite element approximation of the Sobolev constant $S(p,N)$ in the unit ball for $N\ge 2$ and $1<p<N$, establishing the sharp convergence rate $S_h(p,N)-S(p,N) \simeq h^{\alpha(p,N)}$ with $\alpha(p,N)=\tfrac{2(N-p)}{N+p-2}$.A central analytic device is the $p$-Sobolev deficit $\delta(u)$, together with FE-appropriate quasi-norms $|u-v|_{p,2}$, which yield two-sided bounds relating numerical error to the distance to the Sobolev minimizers manifold $\mathcal{M}$.The methodology combines projection of continuous minimizers onto the FE space, refined estimates for minimizers $U_{\lambda,x_0}$, and concentration-compactness to handle non-attainment in bounded domains, producing optimal two-sided bounds that recover the known $p=2$ rate (e.g., $N\ge 3$, $S_h- S\sim h^{\tfrac{2(N-2)}{N}}$) and extend them to general $1<p<N$.These results have implications for sharp FE convergence analyses in nonlinear Sobolev-type inequalities and pave the way for applying the same deficit/quasi-norm framework to related constants.
Abstract
We establish optimal convergence rates for the P1 finite element approximation of the Sobolev constant in arbitrary dimensions N\geq 2 and for Lebesgue exponents 1<p<N. Our analysis relies on a refined study of the Sobolev deficit in suitable quasi-norms, which have been introduced and utilized in the context of finite element approximations of the p- Laplacian. The proof further involves sharp estimates for the finite element approximation of Sobolev minimizers.