Approximated Convex

The goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope. Such a problem arises in various physical situations where the function considered is some energy that has to be minimized.This study is a first step toward understanding how to approximate the quasi-convex envelope of a function. The importance of this issue is due to the various applications that are encountered, in particular, in the field of material science. ©1994 (Copyright) Society for Industrial and Applied Mathematics DOI: 10.1137/0731007 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: http://doi.org/10.5167/uzh-22621 Originally published at: Brighi, B; Chipot, M (1994). Approximated convex envelope of a function. SIAM Journal on Numerical Analysis, 31(1):128-148. DOI: 10.1137/0731007 SIAM J. NUMER. ANAL. Vol. 31, No. 1, pp. 128-148, February 1994 (1994 Society for Industrial and Applied Mathematics 007 APPROXIMATED CONVEX ENVELOPE OF A FUNCTION* BERNARD BRIGHI AND MICHEL CHIPOT Abstract. The goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope. Such a problem arises in various physical situations where the function considered is some energy that has to be minimized. This study is a first step toward understanding how to approximate the quasi-convex envelope of a function. The importance of this issue is due to the various applications that are encountered, in particular, in the field of material science. The goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope. Such a problem arises in various physical situations where the function considered is some energy that has to be minimized. This study is a first step toward understanding how to approximate the quasi-convex envelope of a function. The importance of this issue is due to the various applications that are encountered, in particular, in the field of material science. Key words, nonconvex energy, convex envelope, finite elements, approximation AMS subject classifications. 26B25, 49XX, 52A20, 65N30, 65N99