# American Institute of Mathematical Sciences

September  2015, 35(9): 4041-4070. doi: 10.3934/dcds.2015.35.4041

## Value iteration convergence of $\epsilon$-monotone schemes for stationary Hamilton-Jacobi equations

 1 Laboratoire Jacques-Louis Lions, UMR 7598, Université Paris-Diderot (Paris 7), UFR de Mathématiques - 5 rue Thomas Mann, 75205 Paris CEDEX 13 2 Dipartimento di Matematica, Istituto "Guido Castelnuovo", Sapienza Università di Roma, Piazzale Aldo Moro, 2 I-00185 Roma 3 Dipartimento di Matematica e Fisica, Università di Roma Tre, L.go S. Leonardo Murialdo, 1, 00146 Roma, Italy 4 Mathematisches Institut, Fakultät für Mathematik, Physik und Informatik, Universität Bayreuth, 95440 Bayreuth, Germany 5 Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz, Austria 6 Unité des mathématiques appliquées (UMA), ENSTA ParisTech, 828 Bd Maréchaux, 91120 Palaiseau

Received  April 2014 Published  April 2015

We present an abstract convergence result for the fixed point approximation of stationary Hamilton--Jacobi equations. The basic assumptions on the discrete operator are invariance with respect to the addition of constants, $\epsilon$-monotonicity and consistency. The result can be applied to various high-order approximation schemes which are illustrated in the paper. Several applications to Hamilton--Jacobi equations and numerical tests are presented.
Citation: Olivier Bokanowski, Maurizio Falcone, Roberto Ferretti, Lars Grüne, Dante Kalise, Hasnaa Zidani. Value iteration convergence of $\epsilon$-monotone schemes for stationary Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4041-4070. doi: 10.3934/dcds.2015.35.4041
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