An adaptive finite element method for solving a double well problem describing crystalline microstructure
24098 Kiel, Germany. email@example.com.
Revised: 12 September 1998
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.
Mathematics Subject Classification: 65K10 / 65M50 / 65N30 / 73C50 / 73S10
Key words: Adaptive algorithm / finite element method / nonconvex minimization / multi-well problem / microstructure / multiscale / nonlinear elasticity / shape-memory alloy / materials science.
© EDP Sciences, SMAI, 1999