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

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.

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
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