Geometric local parameterization for solving Hele-Shaw problems with surface tension

In this work, we introduce a novel computational framework for solving the two-dimensional Hele-Shaw free boundary problem with surface tension. The moving boundary is represented by point clouds, eliminating the need for a global parameterization. Our approach leverages Generalized Moving Least Squares (GMLS) to construct local geometric charts, enabling high-order approximations of geometric quantities such as curvature directly from the point cloud data. This local parameterization is systematically employed to discretize the governing boundary integral equation, including an analytical formula of the singular integrals. We provide a rigorous convergence analysis for the proposed spatial discretization, establishing consistency and stability under certain conditions. The resulting error bound is derived in terms of the size of the uniformly sampled point cloud data on the moving boundary, the smoothness of the boundary, and the order of the numerical quadrature rule. Numerical experiments confirm the theoretical findings, demonstrating high-order spatial convergence and the expected temporal convergence rates. The method's effectiveness is further illustrated through simulations of complex initial shapes, which correctly evolve towards circular equilibrium states under the influence of surface tension.