| Abstract: |
| We present a computational framework for solving the two-dimensional Hele Shaw free boundary problem with surface tension using a mesh free point cloud representation of the moving interface. By avoiding global parameterization, the method naturally accommodates complex and evolving geometries. Our approach leverages Generalized Moving Least Squares to construct local geometric charts on the boundary, enabling accurate approximation of geometric quantities such as curvature directly from scattered points. This local structure is used to discretize the boundary integral formulation of the problem, with singular integrals evaluated through analytical expressions to maintain high accuracy. We provide a convergence analysis establishing consistency and stability of the spatial discretization, with error bounds depending on point cloud density, boundary smoothness, and quadrature order. Numerical experiments confirm spatial convergence and demonstrate robust evolution of complex initial interfaces toward circular equilibrium under surface tension. |
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