An optimal method for high order mixed derivatives of bivariate functions

TL;DR

This paper introduces an order-optimal numerical differentiation method for high-order mixed derivatives of bivariate functions with finite smoothness, balancing accuracy and information usage.

Contribution

It presents a truncation-based algorithm that achieves optimal accuracy and efficiency in recovering derivatives of bivariate functions.

Findings

The method is proven to be order-optimal in accuracy.

Numerical examples confirm successful implementation.

The approach effectively balances accuracy and information use.

Abstract

The problem of optimal recovering high-order mixed derivatives of bivariate functions with finite smoothness is studied. Based on the truncation method, an algorithm for numerical differentiation is constructed, which is order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. Numerical examples are provided to illustrate the fact that our approach can be implemented successfully.