Adaptive Approximation of Young Measure Solutions in Scalar Nonconvex Variational Problems

This paper addresses the numerical approximation of Young measures appearing as generalized solutions to scalar nonconvex variational problems. We prove a priori and a posteriori error estimates for a macroscopic quantity, the stress. For a scalar three-well problem we show convergence of other quantities such as Young measure support and microstructure region. Numerical experiments indicate that the computational effort in the solution of the large optimization problem is significantly reduced by using an adaptive mesh refinement strategy based on a posteriori error estimates in combination with an active set strategy due to Carstensen and Roub́ıček [Numer. Math., 84 (2000), pp. 395–414].