Calculus of Variations and Geometric Measure Theory

P. F. Antonietti - A. Pratelli

Finite Element Approximation of the Sobolev Constant

created by pratelli on 30 Sep 2009
modified on 16 Feb 2015


Published Paper

Inserted: 30 sep 2009
Last Updated: 16 feb 2015

Journal: Numer. Math.
Year: 2010


Denoting by $S$ the sharp constant in the Sobolev inequality in ${\rm W}^{1,2}_0(B)$, being $B$ the unit ball in $R^3$, and denoting by $S_h$ its approximation in a suitable finite element space, we show that $S_h$ converges to $S$ as $h$ goes to $0$ 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.