Higher order derivative estimates for finite-difference schemes for linear elliptic and parabolic equations. (English) Zbl 1183.65107

Gyöngy, István; Krylov, Nicolai

Methods Appl. Anal. 16, No. 2, 187-216 (2009).

This paper is concerned with the study of smoothness of solution \(u_h\) for finite difference schemes related to parabolic and elliptic equations given on the whole \(\mathbb{R}^d\).
First, the authors consider a grid in \(\mathbb{R}^d\) and a large class of monotone finite difference schemes in the space variable \(x\). For each small parameter \(h>0\), the given grid is dilated by \(h\) and for each \(x\in\mathbb{R}^d\) it is shifted so that \(x\) becomes a mesh point.
The main results give estimates, independent of \(h\), for any order of the derivatives of \(u_h\) in \(x\). Using these estimates, the authors next provide estimates for the derivative of \(u_h\) in \(h\) and thus develop a new method to expand in power series \(u_h\) in terms of \(h\). Furthermore, under general conditions, it is shown in the paper that the accuracy of finite difference schemes can be improved to any order by taking suitable linear combinations of finite difference approximations with different mesh-sizes.

Reviewer: Marius Ghergu (Dublin)

MSC:

65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
35J70	Degenerate elliptic equations
35K65	Degenerate parabolic equations
35K10	Second-order parabolic equations
35J25	Boundary value problems for second-order elliptic equations
65N06	Finite difference methods for boundary value problems involving PDEs
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs

Keywords:

finite differences approximations; high order derivative estimates; degenerate parabolic problem; elliptic equations; monotone finite difference schemes