First derivatives estimates for finite-difference schemes

Gyöngy, István; Krylov, Nicolai

doi:10.1090/S0025-5718-09-02229-7

First derivatives estimates for finite-difference schemes
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by István Gyöngy and Nicolai Krylov PDF

Math. Comp. 78 (2009), 2019-2046 Request permission

Abstract:

We give sufficient conditions under which solutions of discretized in space second-order parabolic and elliptic equations, perhaps degenerate, admit estimates of the first derivatives in the space variables independent of the mesh size.

References

Hongjie Dong and N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellman equations with constant coefficients, Algebra i Analiz 17 (2005), no. 2, 108–132; English transl., St. Petersburg Math. J. 17 (2006), no. 2, 295–313. MR 2159586, DOI 10.1090/S1061-0022-06-00905-8
Hongjie Dong and Nicolai V. Krylov, Rate of convergence of finite-difference approximations for degenerate linear parabolic equations with $C^1$ and $C^2$ coefficients, Electron. J. Differential Equations (2005), No. 102, 25. MR 2162263
Hongjie Dong and Nicolai V. Krylov, The rate of convergence of finite-difference approximations for parabolic Bellman equations with Lipschitz coefficients in cylindrical domains, Appl. Math. Optim. 56 (2007), no. 1, 37–66. MR 2334605, DOI 10.1007/s00245-007-0879-4
István Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II, Potential Anal. 11 (1999), no. 1, 1–37. MR 1699161, DOI 10.1023/A:1008699504438
I. Gyöngy and N.V. Krylov, Accelerated convergence of finite difference schemes for second order parabolic and elliptic PDEs, in preparation.
N. V. Krylov, A priori estimates of smoothness of solutions to difference Bellman equations with linear and quasi-linear operators, Math. Comp. 76 (2007), no. 258, 669–698. MR 2291833, DOI 10.1090/S0025-5718-07-01953-9
N. V. Krylov, On factorizations of smooth nonnegative matrix-values functions and on smooth functions with values in polyhedra, Appl. Math. Optim. 58 (2008), no. 3, 373–392. MR 2456852, DOI 10.1007/s00245-008-9040-2
Walter Littman, Résolution du problème de Dirichlet par la méthode des différences finies, C. R. Acad. Sci. Paris 247 (1958), 2270–2272 (French). MR 107748
J. Von Neumann and R. D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys. 21 (1950), 232–237. MR 37613, DOI 10.1063/1.1699639
Hyek Yoo, Semi-discretization of stochastic partial differential equations on $\textbf {R}^1$ by a finite-difference method, Math. Comp. 69 (2000), no. 230, 653–666. MR 1654030, DOI 10.1090/S0025-5718-99-01150-3