Numerical analysis of oscillations in multiple well problems. (English) Zbl 0824.65045

Chipot, M.; Collins, C.; Kinderlehrer, D.

Numer. Math. 70, No. 3, 259-282 (1995).

Obviously this is a generalized and refined version of a former investigation of C. Collins, D. Kinderlehrer and M. Luskin [SIAM J. Numer. Anal. 28, No. 2, 321-332 (1991; Zbl 0725.65067)]. For non-convex variational problems, occurring e.g. in the study of ordered materials as crystals, in many cases, the infimum of energy integrals cannot be attained. Thus, the authors focus on the cases where there is no minimizing function (here a Borel function of matrices, vanishing for \(k\) distinct matrices – just this is the problem with \(k\) potential wells).
They state the conditions for the absence of minimizers. The main topic is the determination of the analytic properties of minimizing sequences. Essential tool is the (unique) definition of a parametrized measure, called a Young measure, characterized by D. Kinderlehrer and P. Pedregal [Arch. Rat. Mech. Anal. 115, No. 4, 329-365 (1991; Zbl 0754.49020), C.R. Acad. Sci., Paris, Ser. I 313, No. 11, 765-770 (1991; Zbl 0759.49005)]. Further, a (piecewise linear) finite element method approximates the properties of these sequences (and leads to an approximate value of the infimum). A rotationally invariant two-well problem has been studied (also numerically) as example.

Reviewer: E.Lanckau (Chemnitz)

MSC:

65K10	Numerical optimization and variational techniques
49M15	Newton-type methods
49J40	Variational inequalities

Keywords:

Galerkin methods; error bounds; numerical example; variational methods; oscillations; multiple well problems; non-convex variational problems; Young measure; finite element method