mGFD: A meshless generalized finite difference method

Abstract

This work introduces a novel meshless method, the meshless Generalized Finite Difference (mGFD) scheme, which is derived from an optimization formulation that enforces the consistency condition. This approach eliminates the need for additional weight functions required by other methods, enabling efficient and accurate simulations of complex geometries. The method leverages a flexible, node-based discretization scheme that avoids a predefined mesh, providing enhanced versatility and adaptability in modeling various engineering applications. The proposed method's flexibility and adaptability are demonstrated through numerical solutions of elliptic, parabolic, and hyperbolic partial differential equations in highly irregular domains, providing satisfactory results compared to known exact solutions.