[Paper Review] A Cubic Whitney and Further Developments in Geometric Discretisation

Samik Sen|ArXiv.org|Jul 17, 2003

Advanced Numerical Analysis Techniques7 references3 citations

TL;DR

This paper introduces a cubic Whitney map and a metric-compatible discrete Hodge star in geometric discretisation, ensuring convergence of discrete operators to their continuum counterparts. By modifying the inner product to preserve the duality between the Hodge star and inner product, the authors derive volume factors that guarantee convergence of the codifferential and Laplacian, resolving a key issue in lattice field theory and finite element methods.

ABSTRACT

Geometric discretisation draws analogies between discrete objects and operations on a complex with continuum ones on a manifold. We generalise the theory to the cubic case and incorporate metric, by adding volume factors to our discrete Hodge star and then by modifying our inner product which leads to the same result.

Motivation & Objective

• To generalize geometric discretisation to the cubic case, extending the Whitney map beyond simplicial complexes.
• To resolve the inconsistency between the discrete Hodge star and inner product by introducing a metric-compatible inner product.
• To demonstrate convergence of the discrete codifferential and Laplacian to their continuum analogues using volume factors derived from geometric reasoning.
• To establish a foundation for applications in lattice field theory, particularly in preserving chirality without degeneracy.

Proposed method

• Introduce a cubic Whitney map to extend geometric discretisation from simplicial to cubical complexes.
• Modify the discrete Hodge star by incorporating volume factors derived from the ratio of dual cell volumes to primal cell volumes.
• Define a new inner product that naturally induces the required volume factors, preserving the duality $\star^2 = I$ and $\delta = \star d \star$.
• Use a heuristic based on scaling with lattice spacing $a$ to derive expected convergence behavior for $p$-cochains in $D$ dimensions.
• Verify convergence by comparing discrete operators like $\delta = \star d \star$ against continuum results in 2D and 3D cases.
• Show that the volume factors derived from the heuristic match exactly those induced by the new inner product, ensuring consistency.

Experimental results

Research questions

• RQ1 How can geometric discretisation be extended from simplicial to cubic complexes while preserving topological and metric structures?
• RQ2 What volume factors are necessary for the discrete Hodge star to ensure convergence of the codifferential and Laplacian to their continuum limits?
• RQ3 Can a metric-compatible inner product be defined such that it induces the correct volume factors for the Hodge star without breaking the fundamental identities $\star^2 = I$ and $\delta = \star d \star$?
• RQ4 How do the discrete operators behave in 2D and 3D under the new formulation, and do they converge to the continuum as the lattice spacing $a \to 0$?
• RQ5 What is the relationship between the heuristic volume factors and the inner product-induced factors in the context of geometric discretisation?

Key findings

• The cubic Whitney map successfully extends geometric discretisation to cubical complexes, maintaining topological consistency.
• Volume factors of the form $a^{D-2p}$ for $p$-cochains in $D$ dimensions are required for convergence of the discrete Hodge star and associated operators.
• The heuristic derivation of volume factors based on cell volume ratios exactly matches the factors induced by a modified inner product, resolving a prior inconsistency.
• The new inner product preserves the fundamental identity $\delta = \star d \star$ while ensuring convergence of the codifferential and Laplacian to their continuum counterparts.
• In 2D, the factors are $a^2$ for vertices, $1$ for edges, and $a^{-2}$ for faces; in 3D, they are $a^3$, $a$, $a^{-1}$, and $a^{-3}$, respectively.
• The convergence of discrete operators is confirmed by explicit comparison with continuum results in 2D, showing agreement in the $a \to 0$ limit.