A unified CPM framework using the least-squares generalized finite difference method for surface PDEs

This paper introduces a novel framework, the unified closest point method (CPM) using the least-squares generalized finite difference method (LS-GFDM), to solve the surface partial differential equations (PDEs). Our approach addresses key limitations of existing embedding methods by unifying the computation of finite difference weights and interpolation into a single step, enabling efficient and reasonably accurate solutions on scattered data points. Unlike traditional CPM frameworks that rely on the finite difference method (FDM) and require interpolation to restrict solutions on the surface, potentially introducing additional errors, our LS-GFDM eliminates the need for interpolation. Furthermore, compared to the orthogonal gradient method (OrG) based on radial basis function (RBF) methods, which requires specialized data point arrangements, our method offers greater flexibility by accommodating any arbitrary point set. Higher-order schemes for LS-GFDM are easily achieved by increasing the order of the Taylor series expansion, and the method can be readily extended to other embedded methods. Numerical experiments demonstrate that the proposed framework is less sensitive to the narrow band domain size and showcases good effectiveness and robustness for solving surface PDEs.