Convergent finite elements for a class of nonconvex variational problems
Bernd Kawohl · IMA Journal of Numerical Analysis · 1998
We study the finite element discretization of the abstract minimization problem min{F(u)}. The functional F is neither convex nor growing at ∞. For the admissible class CM = {u : Ω → R, u concave, 0 ⩽ u(x) ⩽ M} polygonal domains Ω ⊂ R2 and linear Courant triangles, we show the convergence of the finite element approximations to a minimizer of F(u). A class of projected Newton methods for the discrete problems yields locally super-linear convergence. We present numerical experiments for a model functional F, related to Newton's problem of minimal resistance of a body moving through a fluid.