Meshfree Methods on Manifolds for Hydrodynamic Flows on Curved Surfaces: A Generalized Moving Least-Squares (GMLS) Approach
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We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and a splitting scheme of the fourth-order governing equations in terms of two second-order elliptic operators. We show our methods have high-order convergence rates for the metric and other geometric quantities of the manifold, for the truncation errors of exterior calculus operators, and for the solution errors of the Stokes problem for hydrodynamic flows on curved surfaces. Our approaches also may be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.
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