zbMATH — the first resource for mathematics

Approximation in nonconvex problems. (English) Zbl 0793.41027
Bandle, C. (ed.) et al., Progress in partial differential equations: calculus of variations, applications. 1st European conference on Elliptic and parabolic problems, Pont-à-Mousson, France, June 1991. Harlow, Essex: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 267, 150-157 (1993).
Let $$\oint:\mathbb{R}^ n \to \mathbb{R}$$ be a continuous function, $$\oint^{**}$$ be the convex envelope of $$\oint$$ i.e. the function defined by $$\oint^{**} (\alpha)=\sup \{g(\alpha) : g$$ convex, $$g \leq \oint\}$$, $$\Omega$$ be some bounded, polygonal domain of $$\mathbb{R}^ n$$, $$n \geq 1$$, with a boundary $$\Gamma$$, $$\tau$$ denotes a triangulation of $$\Omega$$ into simplifices $$K$$ of mesh size $$h=\text{Max}_{K \in \tau} \text{diam} K$$, $$P_ 1(K)$$ denotes the set of polynomials of degree 1 on $$K$$ and $$V^ 0_ h=\{v:\Omega \to \mathbb{R}$$: $$v$$ is continuous, $$v |_ K \in P_ 1(K)$$ $$\forall K \in \tau$$, $$v=0$$ on $$\Gamma\}$$ ($$v |_ K$$ denotes the restriction of $$v$$ on $$K)$$. Define the approximated convex envelope of $$\oint$$ by $\oint^{**}_ h(\alpha)=\inf_{v \in V^ 0_ h} \left\{ {1 \over \Omega} \int_ \Omega \oint (\alpha+\nabla v(x)) dx \right\}.$ clearly $$\oint^{**}_ h(\alpha) \geq \oint^{**} (\alpha)$$. In this note, estimates for the difference $$\oint^{**}_ h- \oint^{**}$$ are derived in terms of the mesh size $$h$$. Restricting to the one dimensional case, the following theorem is proved: Assume that $$\lim_{| \xi | \to \infty} {\oint (\xi) \over | \xi |}=+\infty$$, $$\Omega=(w_ -,w_ +)$$. Denote by $$x_ 0,x_ 1,\dots,x_{n+1}$$ a subdivision of $$\Omega$$ such that $$w_ -=x_ 0, x_ 1, \dots,x_ n$$, $$x_{n+1}=w_ +$$ and $$h=\max_{i=0,\dots,n} x_{i+1}-x_ i$$. Then there exists a constant $$c=c(\alpha,\oint)$$ such that $0 \leq \oint_ h^{**} (\alpha)-\oint^{**} (\alpha) \leq {c \over | \Omega |} \cdot h$ where $$| \Omega |=w_ =-w_ -$$.
For the entire collection see [Zbl 0780.00014].

MSC:
 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)