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New numerical methods are keeping to emerge due to increasing needs in science and engineering. In this context, a novel mesh-meshless duality principle is proposed to build new numerical algorithms in this paper. The basic principle is that the approximations of unknown functions and their derivatives and the discretization scheme, could use mesh- or meshless- techniques distinctly, which results in the simultaneous use of the mesh and the meshless aspects of the same nodes while not bringing more difficulties in processing grid topology. The proposed local integral-generalized finite difference (IGFD) method is a specific implementation of this principle. For IGFD method, the approximations of unknown functions and their derivatives are realized by combining Taylor series expansion on local scatter nodes and least square method. The discretization of partial differential equations (PDEs) is based on the local weak form on the boundaries of local domain, which ensures the local conservation. The proposed method inherits the high accuracy of GFD method, and it is easier to deal with various boundary conditions. Some numerical examples show that this novel method demonstrates sound accuracy and is suitable for solving many PDE problems. It has sufficient potential for further development in solving more complex issues. [ABSTRACT FROM AUTHOR]