# Meshfree Methods on Manifolds for Hydrodynamic Flows on Curved Surfaces: A Generalized Moving Least-Squares (GMLS) Approach

## TL;DR

This paper introduces a meshfree GMLS-based method for simulating hydrodynamic flows on curved manifolds, leveraging exterior calculus for high-order accuracy and geometric computations.

## Contribution

It develops a generalized moving least squares approach combined with exterior calculus to efficiently solve hydrodynamic equations on manifolds with high-order convergence.

## Key findings

- High-order convergence demonstrated for flows on curved surfaces
- Accurate approximation of geometric quantities like metric and curvature
- Applicable to scalar and vector problems on manifolds

## Abstract

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 splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.