Approximation Schemes for Monotone Systems of Nonlinear Second Order Partial Differential Equations: Convergence Result and Error Estimate

Ariela Briani (1, 2) , Fabio Camilli (3) , Hasnaa Zidani (4, 2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 OC - Optimisation et commande
3 MeMoMat - Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate
4 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems

We consider approximation schemes for monotone systems of fully nonlinear second order partial di erential equations. We rst prove a general convergence result for monotone, consistent and regular schemes. This result is a generalization to the well known framework of Barles-Souganidis, in the case of scalar nonlinear equation. Our second main result provides the convergence rate of approximation schemes for weakly coupled systems of Hamilton-Jacobi-Bellman equations. Examples including nite di erence schemes and Semi-Lagrangian schemes are discussed.

Mots clés: Monotone systems; viscosity solution; approximation scheme; error estimate; error estimate.

Domaines: Optimisation et contrôle [math.OC]

Ariela Briani, Fabio Camilli, Hasnaa Zidani. Approximation Schemes for Monotone Systems of Nonlinear Second Order Partial Differential Equations: Convergence Result and Error Estimate. Differential Equations and Applications, 2012, 4, pp.297-317. ⟨10.7153/dea-04-18⟩. ⟨inria-00627520⟩