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.

翻译：本文提出了一种新颖的计算框架，用于求解具有表面张力的二维Hele-Shaw自由边界问题。该框架采用点云表示移动边界，从而无需全局参数化。我们利用广义移动最小二乘法构建局部几何图册，能够直接从点云数据对曲率等几何量进行高阶逼近。这种局部参数化方法被系统地用于离散化控制边界积分方程，其中包括奇异积分的解析公式。我们对所提出的空间离散化方案进行了严格的收敛性分析，在一定条件下建立了一致性和稳定性。所得误差界是根据移动边界上均匀采样点云数据的尺寸、边界的平滑度以及数值积分法则的阶数推导得出的。数值实验验证了理论结果，展示了高阶空间收敛性和预期的时序收敛率。通过模拟复杂初始形状在表面张力作用下正确演化为圆形平衡态的过程，进一步证明了该方法的有效性。