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Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations

Authors: Guy Barles and Espen R. Jakobsen
Journal: Math. Comp. 76 (2007), 1861-1893
MSC (2000): Primary 65M15, 65M06, 35K60, 35K70, 49L25
Published electronically: April 20, 2007
MathSciNet review: 2336272
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Abstract: We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton-Jacobi-Bellman equations by introducing a new notion of consistency. Our results are robust and general - they improve and extend earlier results by Krylov, Barles, and Jakobsen. We apply our general results to various schemes including Crank-Nicholson type finite difference schemes, splitting methods, and the classical approximation by piecewise constant controls. In the first two cases our results are new, and in the last two cases the results are obtained by a new method which we develop here.

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Additional Information

Guy Barles
Affiliation: Laboratoire de Mathématiques et Physique Théorique, University of Tours, 37200 Tours, France

Espen R. Jakobsen
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway

Keywords: Hamilton-Jacobi-Bellman equations, switching system, viscosity solution, approximation schemes, finite difference methods, splitting methods, convergence rate, error bound
Received by editor(s): June 24, 2005
Received by editor(s) in revised form: June 29, 2006
Published electronically: April 20, 2007
Additional Notes: Jakobsen was supported by the Research Council of Norway, grant no. 151608/432
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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