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Numerical Methods for PDEs on Surfaces

Abstract
We present recent numerical methods for solving partial differential equations on manifolds and point clouds. In the first part of the talk, we introduce both the grid-based particle method (GBPM) and the cell-based particle method (CBPM) for dynamic interface problems. Then, we discuss new and simple discretization, named the Modified Virtual Grid Difference (MVGD), for the numerical approximation of the Laplace-Beltrami on manifolds sampled by point clouds. We first introduce a local virtual with a scale adapted to the sampling density centered at each point. Then propose a modified finite difference scheme on the virtual grid to discretize the LB operator. The new discretization provides more diagonal dominance to the resulting linear system and improves its conditioning. In the last part, we present a local regularized least squares radial basis function (RLS-RBF) method for solving partial differential equations on irregular domains or manifolds. The idea extends the standard RBF method by replacing the interpolation in the reconstruction with the least-squares fitting approximation.