An adaptive finite element method for solving a double well problem describing crystalline microstructure

Mathematisches Seminar, Christian-Albrechts-Universität Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany. apr@numerik.uni-kiel.de.

Received: 27 February 1998
Revised: 12 September 1998

Abstract

The minimization of nonconvex functionals naturally arises in materials sciences where deformation gradients in certain alloys exhibit microstructures. For example, minimizing sequences of the nonconvex Ericksen-James energy can be associated with deformations in martensitic materials that are observed in experiments[2,3]. — From the numerical point of view, classical conforming and nonconforming finite element discretizations have been observed to give minimizers with their quality being highly dependent on the underlying triangulation, see [8,24,26,27] for a survey. Recently, a new approach has been proposed and analyzed in [15,16] that is based on discontinuous finite elements to reduce the pollution effect of a general triangulation on the computed minimizer. The goal of the present paper is to propose and analyze an adaptive method, giving a more accurate resolution of laminated microstructure on arbitrary grids.