Finite Element Approximation of the Sobolev Constant

Code:
29/2009
Title:
Finite Element Approximation of the Sobolev Constant
Date:
Wednesday 21st October 2009
Author(s):
Antonietti, Paola F.; Pratelli, Aldo
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Abstract:
Denoting by $S$ the sharp constant in the Sobolev inequality in $W^{1,2}_0(B)$, being $B$ the unit ball in $R3$, and denoting by $S_h$ its approximation in a suitable finite element space, we show that $S_h$ converges to $S$ as $h->0$ 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.