Numerical analysis of the (relaxed) double well problem

The mathematical modelling of phase transitions in alloys, or optimal design of composites, is concerned with non-convex variational problems, where a minimum may or may not be attained. Young's model example, the two-well or double-well problem (P), is considered in this talk. Convexification yields a relaxed problem (RP) related to (P). The loss of information we face in recasting (P) as (RP) is not substantial as long as we are interested in the global displacements, stress fields and Young measures (which describe oscillations creating microstructures); these variables defined by (P) can be computed from solutions of (RP).
Typically, the problem (RP) is convex, but not strictly convex. Hence error estimates for finite element methods are more difficult to prove than for simpler uniformly convex problems. A priori and a posteriori error estimates are presented and illustrated in numerical examples which indicate that it is more reasonable to perform a numerical analysis of (RP) rather than of (P).