Eulerian Geometric Discretizations of Manifolds and Dynamics

, 2012

Category: Scientific Computing

Overall Rating

4.0/5 (28/35 pts)

This paper provides a strong theoretical foundation for building numerical methods that inherently preserve geometric structures using Discrete Exterior Calculus.
While the specific implementations proposed face practical limitations on complex domains and competition from more general modern methods, the core concept of leveraging discrete geometric properties (like orthogonal duals and discrete differential operators) to construct stable and conservative systems remains a valuable insight.
This offers a unique path for designing learned physics models whose architecture is constrained by underlying geometric principles, rather than just learning approximations of existing numerical schemes.

This thesis presents a comprehensive framework for structure-preserving numerical methods on discrete manifolds (meshes), specifically in an Eulerian setting.
Specifically, Chapter 4's derivation of structure-preserving Eulerian integrators via discrete volume-preserving diffeomorphisms and geometric mechanics offers a unique perspective on the configuration space and dynamics of discrete systems on meshes.
This means the model wouldn't just predict motion; it would predict structure-preserving motion.
Furthermore, Chapter 3's work on Hodge-optimized meshes using Optimal Transport provides a method to generate the underlying geometric substrate itself, ensuring that the discrete operators used by the learned simulator (or for traditional simulations) are as accurate as possible...

The focus on "regular grids" for Lie advection (Chapter 2) is a significant limitation in a world increasingly reliant on complex, unstructured, or adaptive meshes for real-world geometry and phenomena.
The requirement for orthogonal primal-dual meshes... is a major bottleneck.
The Lie advection method (Chapter 2) lacks formal error analysis for arbitrary forms and relies on a dimension splitting whose theoretical justification for higher dimensions and non-scalar forms is questionable.
More general frameworks like Finite Element Exterior Calculus (FEEC)... offer robust methods for discretizing differential forms and operators on arbitrary simplicial meshes without requiring restrictive orthogonal dualities or specialized mesh generation.

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