# Higher order nonlocal operator method

## TL;DR

This paper introduces a higher order nonlocal operator method that enhances the original approach by achieving higher convergence rates and simultaneously obtaining multiple derivatives without shape functions, demonstrated through various numerical examples.

## Contribution

The paper extends the nonlocal operator method to higher order schemes using Taylor series, improving convergence and derivative computation without shape functions.

## Key findings

- Achieves higher order convergence compared to the original method.
- Simultaneously obtains multiple derivatives up to a specified order.
- Demonstrates effectiveness through numerical examples in strong and weak forms.

## Abstract

We extend the nonlocal operator method to higher order scheme by using a higher order Taylor series expansion of the unknown field. Such a higher order scheme improves the original nonlocal operator method proposed by the authors in [A nonlocal operator method for solving partial differential equations], which can only achieve one-order convergence. The higher order nonlocal operator method obtains all partial derivatives with specified maximal order simultaneously without resorting to shape functions. The functional based on the nonlocal operators converts the construction of residual and stiffness matrix into a series of matrix multiplication on the nonlocal operator matrix. Several numerical examples solved by strong form or weak form are presented to show the capabilities of this method.