Higher order nonlocal operator method
For researchers in computational PDEs, this improves accuracy of the nonlocal operator method, which previously was limited to first-order convergence.
The authors extend the nonlocal operator method to higher order using Taylor series expansion, achieving convergence beyond first order. Numerical examples demonstrate the method's capabilities in both strong and weak forms.
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.