Numerical analysis of compatible phase transitions in elastic solids

The variational model of phase transitions for elastic materials based on l inearized elasticity leads to a nonconvex minimization problem (P) in which a minimum need not be attained. In the design of advanced materials, the m ain interest is in reliable numerical predictions of certain macroscopic qu antities 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 p roblem. First, a mathematical a priori and a posteriori error analysis is e stablished for the finite element approximation of the stress variable; the n the residual based error indicator is implemented within an adaptive mesh -refinement algorithm. Numerical examples illustrate that the macroscopic p roperties of the materials are computed efficiently with appropriate error control.