Abstract
A numerical method for a (possibly nonconvex) scalar variational problem for the functional to be minimized where u ∈ W 1, p (Ω) and u|∂Ω = u D is proposed; Ω ⊂ ℝ 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 ϕ1(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 Ω 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.
Published Version
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