Numerical solution of the scalar double-well problem allowing microstructure

The direct numerical solution of a non-convex variational problem ($P$) typically faces the difficulty of the finite element approximation of rapid oscillations. Although the oscillatory discrete minimisers are properly related to corresponding Young measures and describe real physical phenomena, they are costly and difficult to compute. In this work, we treat the scalar double-well problem by numerical solution of the relaxed problem ($RP$) leading to a (degenerate) convex minimisation problem. The problem ($RP$) has a minimiser $u$ and a related stress field $\sigma = DW^{**}(\nabla {u})$ which is known to coincide with the stress field obtained by solving ($P$) in a generalised sense involving Young measures. If $u_h$ is a finite element solution, $\sigma _h:= D W^{**}(\nabla {u}_h)$ is the related discrete stress field. We prove a priori and a posteriori estimates for $\sigma -\sigma _h$ in $L^{4/3}(\Omega )$ and weaker weighted estimates for $\nabla {u}-\nabla {u}_h$. The a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments.