Finite element approximation of the Sobolev constant (Q621313)

Let \(W^{1,2}({\mathbb R}^3)\) denote the Sobolev space of all functions \(f\in L^2({\mathbb R}^3)\) whose weak derivative \(Df\) is in \( L^2({\mathbb R}^3)\) too. Then the Sobolev inequality says that \[ \|Df\|_{L^2({\mathbb R}^3)} \geq S\, \|f\|_{L^6({\mathbb R}^3)} \] for any \(f\in W^{1,2}({\mathbb R}^3)\), where \(S\) is the best possible Sobolev constant. Let \(B\subset {\mathbb R}^3\) be the unit ball. Let \(V_h\) be a set of finite element functions of \(B\) which are continuously, piecewise linearly and vanishing on the boundary of \(B\). Here \(h>0\) is the mesh size. Since \(V_h \subset W^{1,2}({\mathbb R}^3)\), there exists a best possible discrete Sobolev constant \(S_h\) with \[ \|Df\|_{L^2(B)} \geq S_h\, \|f\|_{L^6(B)} \] for all \(f\in V_h\). The authors show that \(S + \frac{1}{C}\, h^{\gamma} \leq S_h \leq S + C\,h^{1/3}\) for constants \(C,\, \gamma >0\) such that \(S_h \to S\) for \(h\to 0\). Numerical results including an adaptive refinement strategy are presented too.