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 $u\mathrm{\_}h$ discretized with $P\mathrm{\_}1$ Finite Elements, under the constraint that $u\mathrm{\_}h$ be convex, cannot converge. Here, we first improve this result by proving that non-convergence is due to the mesh refinment lack of richness, remains local and is true even for any mesh. Then, we investigate the consistency of various natural discretizations ($P\mathrm{\_}1$ and $P\mathrm{\_}2$) of second order constraints (subharmonicity and convexity) without discussing the convergence. We also numerically illustrate convergence of a method proposed in the literature that is simpler than existing methods.