Numerical analysis of compatible phase transitions in elastic solids

The variational model of phase transitions for elastic materials based on linearized elasticity leads to a nonconvex minimization problem (P) in which a minimum need not be attained. In the design of advanced materials, the main interest is in reliable numerical predictions of certain macroscopic quantities such as the global deformation and the stress field determined in a relaxed problem (QP). An explicit formula of the quasi-convexified energy density in (QP) due to R.V. Kohn provides us with a well-posed numerical problem. First, a mathematical a priori and a posteriori error analysis is established for the finite element approximation of the stress variable; then the residual based error indicator is implemented within an adaptive mesh-refinement algorithm. Numerical examples illustrate that the macroscopic properties of the materials are computed efficiently with appropriate error control.