Coarse-convex-compactification approach to numerical solution of nonconvex variational problems

A numerical method for a (possibly nonconvex) scalar variational problem for the functional [image omitted] to be minimized where uW1, p() and u|=uD is proposed; n is a bounded Lipschitz domain, n=1 or 2. This method allows the computation of the Young-measure solution of the generalized relaxed version of the original problem and applies to those cases in which 1(x, ) is polynomial. The Young measures involved in the relaxed problem can be represented by their algebraic moments, and a finite-element mesh is used to discretize and thus to approximate both u and the Young measure (in the momentum representation). Eventually, this obtained convex semidefinite program is solved by efficient specialized mathematical-programming solvers. This method is justified by convergence analysis and eventually tested on a 2-dimensional benchmark numerical example. It serves as an example of how convex compactification can efficiently be used numerically if small enough, that is, coarse enough.