On the numerical analysis of non-convex variational problems

Summary.

We discuss a numerical method for finding Young-measure-valued minimizers of non-convex variational problems. To have any hope of a convergence theorem, one must work in a setting where the minimizer is unique and minimizing sequences converge strongly. This paper has two main goals: (i) we specify a method for producing strongly-convergent minimizing sequences, despite the failure of strict convexity; and (ii) we show how uniqueness of the Young measure can be parlayed into a numerical convergence theorem. The treatment of (ii) is done in the setting of two model problems, one involving scalar valued functions and a multiwell energy, the other from micromagnetics.