Preprint 2003-009

Error Bounds for Monotone Approximation Schemes for Non-Convex Degenerate Elliptic Equations in R₁

Abstract: In this paper we provide estimates of the rates of convergence of monotone approximation schemes for non-convex equations in one space-dimension. The equations under consideration are the degenerate elliptic Isaacs equations with $x$-depending coefficients,and the results applies in particular to finite difference methods and control schemes based on the dynamic programming principle. Recently, Krylov, Barles, and Jakobsen obtained similar estimates for convex Hamilton-Jacobi-Bellman equations in arbitrary space-dimensions. Our results extend these to non-convex equations in one space-dimension and are the first results for non-convex second order equations. Furthermore for finite difference equations, we obtain better rates that Krylov and can handle more general equations than Barles and Jakobsen.

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Author(s):

Espen R. Jakobsen, <erj@math.ntnu.no>

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<erj@math.ntnu.no> January 29 2003.