Optimization under second order constraints: are the finite element discretizations consistent ?

It is proved in Chon\'e and Le Meur (2001) that the problem of minimizing a Dirichlet-like functional of the function $u_h$ discretized with $P_1$ Finite Elements, under the constraint that $u_h$ be convex, cannot converge to the right solution at least on a very wide range of meshes. In this article, we first improve this result by proving that non-convergence is due to a geometrical obstruction and remains local. Then, we investigate the consistency of various natural discretizations ($P_1$ and $P_2$) of second order constraints (subharmonicity and convexity). We also discuss various other methods that have been proposed in the literature.