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

2013/09/05 by Meur, Hervé Le
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1309.1433

Abstract

It is proved in Choné 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. 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_1 and P_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.

Related

• On convex functions and the finite element method
• Enforcing Neumann Boundary Conditions with Polynomial Extension Operators to Acheive Optimal Convergence Rates on Polytopial Meshes in the Finite Element Method
• Nonconforming discretizations of convex minimization problems and precise relations to mixed methods
• First and Second Order Shape Optimization based on Restricted Mesh\n Deformations
• Local Error Estimates of the Finite Element Method for an Elliptic Problem with a Dirac Source Term