A local weak form of the generalized finite difference method (GFDM) with control volume in heat conduction problems

An explicit local weak form of the Generalized Finite Difference Method (GFDM) is developed for solving heat conduction problems by incorporating the concept of control volumes from the Finite Volume Method (FVM). The control volume is introduced as the local integral domain in the local weak form, where the Heaviside function is employed as the weight function, similar to traditional FVM. However, instead of the cell-centered control volume used in FVM, a vertex-centered control volume is applied for integration. The trial function is constructed using a Generalized Finite Difference approximation based on a Taylor expansion. By applying the divergence theorem, the integral of the governing equation over the control volume is transformed into a boundary integral, thereby reducing the continuity requirements for both the trial function and thermal conductivity. Several numerical examples demonstrate the proposed method's accuracy, stability, and convergence. Additionally, it offers local integration scheme without requiring any background grid.