Appl Math Optim (2014) 69:431–458
On the Rate of Convergence of Difference
Approximations for Uniformly Nondegenerate Elliptic
Published online: 5 November 2013
© Springer Science+Business Media New York 2013
Abstract We show that the rate of convergence of solutions of ﬁnite-difference ap-
proximations for uniformly elliptic Bellman’s equations is of order at least h
where h is the mesh size. The equations are considered in smooth bounded domains.
Keywords Fully nonlinear elliptic equations · Bellman’s equations · Finite
The convergence of and error estimates for monotone and consistent approximations
to fully nonlinear, ﬁrst-order PDEs were established a while ago by Crandall and
Lions  and Souganidis .
The convergence of monotone and consistent approximations for fully nonlinear,
possibly degenerate second-order PDEs was ﬁrst proved in Barles and Souganidis
. In a series of papers Kuo and Trudinger [22, 23, 25] also looked in great detail
at the issues of regularity and existence of such approximations for uniformly elliptic
There is also a probabilistic part of the story related to controlled diffusion pro-
cesses, which started long before see Kushner , Kushner and Dupuis , also
see Pragarauskas .
However, in the above cited articles apart from [5, 29], related to the ﬁrst-order
equations, no rate of convergence was established. One can read more about the
past development of the subject in Barles and Jakobsen  and the joint article of
Hongjie Dong and the author . We are going to discuss only some results concern-
ing second-order Bellman’s equations, which arise in many areas of mathematics
The author was partially supported by NSF Grant DMS-1160569.
N.V. Krylov (
University of Minnesota, 127 Vincent Hall, Minneapolis, MN, 55455, USA