A no-go theorem establishes that density-independent mass matrices cause sharp O(h²) energy residuals in discrete vector-invariant compressible Navier-Stokes on Delaunay-Voronoi meshes; the density-weighted construction eliminates the residual and yields global well-posedness plus Lyapunov stability
Discrete Exterior Calculus
11 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allows us to address the various interactions between forms and vector fields (such as Lie derivatives) which are important in applications. Previous attempts at discrete exterior calculus have addressed only differential forms. We also introduce the notion of a circumcentric dual of a simplicial complex. The importance of dual complexes in this field has been well understood, but previous researchers have used barycentric subdivision or barycentric duals. We show that the use of circumcentric duals is crucial in arriving at a theory of discrete exterior calculus that admits both vector fields and forms.
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TNOs lift neural operators to topological cell complexes via Discrete Exterior Calculus for cross-dimensional coupling, subsuming prior NOs and showing accuracy gains on PDE benchmarks with irregular geometries.
Cellular Sheaf Neural Operators use cell complexes, learned restriction maps, and structure-aware message passing to create discretization-aware neural surrogates that preserve constraints in multiphysics PDEs such as MHD.
An active porous media model for cilia-driven duct flow identifies ciliary confinement ratio and mean ciliary fraction as key controls, yielding a linear flow-pressure trade-off.
Establishes equivalence of DEC cochains with generalized Whitney forms to prove convergence rates for the Hodge-Laplacian in full k-form generality on well-centered meshes.
Generalized Yee methods extend Yee's finite-difference scheme to de Rham finite elements while preserving symplecticity under sparse mass-matrix approximations and enabling symplectic particle-in-cell coupling.
A variational framework using differential forms for general dissipative fluids ensures energy conservation, positive entropy production, and encompasses models such as multi-species MHD with dissipation.
A tensor manifold unifies graph topology and vector embeddings for linear-complexity AI-native academic literature retrieval with temporal diffusion and Riemannian indexing.
A discrete exterior calculus formulation of linear elasticity on cell complexes with displacements as primal 0-cochains and validation on classical analytic problems.
A formula is derived relating Wilson loop correlations in Potts lattice gauge theory to a topological quantity in the plaquette random cluster model, enabling proofs about correlation lengths on Z^d at various temperatures and boundary conditions.
Mesh Field Theory proves mesh-based continuum physics reduces to port-Hamiltonian dynamics with topology fixing interconnections and metrics entering only via constitutive relations, enabling MeshFT-Net for stable, data-efficient simulation.
citing papers explorer
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A no-go theorem and its resolution for the discrete compressible barotropic Navier--Stokes equations
A no-go theorem establishes that density-independent mass matrices cause sharp O(h²) energy residuals in discrete vector-invariant compressible Navier-Stokes on Delaunay-Voronoi meshes; the density-weighted construction eliminates the residual and yields global well-posedness plus Lyapunov stability
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Topological Neural Operators
TNOs lift neural operators to topological cell complexes via Discrete Exterior Calculus for cross-dimensional coupling, subsuming prior NOs and showing accuracy gains on PDE benchmarks with irregular geometries.
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Cellular Sheaf Neural Operators for Structure-Preserving Surrogate Modeling of Constrained PDEs
Cellular Sheaf Neural Operators use cell complexes, learned restriction maps, and structure-aware message passing to create discretization-aware neural surrogates that preserve constraints in multiphysics PDEs such as MHD.
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Cilia-driven transport in confined ducts: an active porous media model
An active porous media model for cilia-driven duct flow identifies ciliary confinement ratio and mean ciliary fraction as key controls, yielding a linear flow-pressure trade-off.
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A Framework for Analysis of DEC Approximations to Hodge-Laplacian Problems using Generalized Whitney Forms
Establishes equivalence of DEC cochains with generalized Whitney forms to prove convergence rates for the Hodge-Laplacian in full k-form generality on well-centered meshes.
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Generalized Yee methods: Scalable symplectic finite element Maxwell solvers
Generalized Yee methods extend Yee's finite-difference scheme to de Rham finite elements while preserving symplecticity under sparse mass-matrix approximations and enabling symplectic particle-in-cell coupling.
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Variational formulation of a general dissipative fluid system with differential forms
A variational framework using differential forms for general dissipative fluids ensures energy conservation, positive entropy production, and encompasses models such as multi-species MHD with dissipation.
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Tensor Manifold-Based Graph-Vector Fusion for AI-Native Academic Literature Retrieval
A tensor manifold unifies graph topology and vector embeddings for linear-complexity AI-native academic literature retrieval with temporal diffusion and Riemannian indexing.
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A Geometric Formulation of Linear Elasticity Based on Discrete Exterior Calculus
A discrete exterior calculus formulation of linear elasticity on cell complexes with displacements as primal 0-cochains and validation on classical analytic problems.
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A Topological Formula for Potts Lattice Gauge Theory Correlations
A formula is derived relating Wilson loop correlations in Potts lattice gauge theory to a topological quantity in the plaquette random cluster model, enabling proofs about correlation lengths on Z^d at various temperatures and boundary conditions.
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Mesh Field Theory: Port-Hamiltonian Formulation of Mesh-Based Physics
Mesh Field Theory proves mesh-based continuum physics reduces to port-Hamiltonian dynamics with topology fixing interconnections and metrics entering only via constitutive relations, enabling MeshFT-Net for stable, data-efficient simulation.