A nonlocal operator method for solving partial differential equations

Abstract: We propose a nonlocal operator method for solving partial differential equations (PDEs). The nonlocal operator is derived from the Taylor series expansion of the unknown field, and can be regarded as the integral form "equivalent" to the differential form in the sense of nonlocal interaction. The variation of a nonlocal operator is similar to the derivative of shape function in meshless and finite element methods, thus circumvents difficulty in the calculation of shape function and its derivatives. {The nonlocal operator method is consistent with the variational principle and the weighted residual method, based on which the residual and the tangent stiffness matrix can be obtained with ease.} The nonlocal operator method is equipped with an hourglass energy functional to satisfy the linear consistency of the field. Higher order nonlocal operators and higher order hourglass energy functional are generalized. The functional based on the nonlocal operator converts the construction of residual and stiffness matrix into a series of matrix multiplications on the nonlocal operators. The nonlocal strong forms of different functionals can be obtained easily via support and dual-support, two basic concepts introduced in the paper. Several numerical examples are presented to validate the method.