Finite difference schemes for stochastic partial differential equations in Sobolev spaces

TL;DR

This paper develops $L_p$-estimates for finite difference schemes solving parabolic SPDEs in Sobolev spaces, enabling improved approximation accuracy through Richardson's method.

Contribution

It provides new $L_p$-estimates for finite difference schemes for degenerate parabolic SPDEs in Sobolev spaces, facilitating error expansion and acceleration techniques.

Findings

Derived $L_p$-estimates for finite difference schemes

Established asymptotic error expansion

Enabled Richardson's acceleration of approximations

Abstract

We discuss $L_p$-estimates for finite difference schemes approximating parabolic, possibly degenerate, SPDEs, with initial conditions from $W^m_p$ and free terms taking values in $W^m_p.$ Consequences of these estimates include an asymptotic expansion of the error, allowing the acceleration of the approximation by Richardson's method.