First derivatives estimates for finite-difference schemes

TL;DR

This paper establishes conditions ensuring that solutions to discretized second-order parabolic and elliptic equations have bounded first derivatives, independent of the mesh size, which is crucial for numerical stability and accuracy.

Contribution

It provides new sufficient conditions for first derivative estimates in finite-difference schemes for possibly degenerate second-order PDEs.

Findings

First derivative estimates are achievable under specified conditions.

Estimates are independent of mesh size, enhancing numerical robustness.

Applicable to degenerate parabolic and elliptic equations.

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.