Optimal convergence rates for the finite element approximation of the Sobolev constant
For numerical analysts and researchers in finite element methods, this provides rigorous convergence guarantees for approximating a fundamental constant in Sobolev spaces.
The paper proves optimal convergence rates for P1 finite element approximation of the Sobolev constant in arbitrary dimensions N≥2 and for 1<p<N, using refined Sobolev deficit analysis and sharp estimates for minimizer approximations.
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.