Various communities of researchers have thought about this class of problems; there are results scattered through journals on image processing, geometric mechanics, numerical analysis, graph theory, machine learning, general relativity (and more?) Besides its use in various domains of application, discrete geometry also provides fresh insights into classical, continuous geometry (e.g. Regge’s discussion of Bianchi identities in his original paper on Regge calculus).
Discrete exterior calculus, cohomology, etc.
- Discrete exterior calculus by Desbrun, Hirani, Leok, Marsden
- Discrete Poincare lemma by Desbrun, Leok, Marsden
- A Discrete Theory of Connections on Principal Bundles by Leok, Marsden, Weinstein.
- General Relativity Without Coordinates by Regge (1961) (Introduces Regge calculus.)
- The geometry of classical Regge calculus by Barrett (1987)
- Torsion Degrees of Freedom in the Regge Calculus as Dislocations on the Simplicial Lattice by Schmidt, Kohler (2001)
Discrete spectral geometry
- References on graph Laplacians, etc.
Convergence of discrete geometric quantities to their continuous counterparts.
- Add references.
Discrete geometric mechanics, discrete field theory
- Discrete Euler-Poincaré and Lie-Poisson Equations by Marsden, Pekarsky, Shkoller (1999)
- Multisymplectic geometry, variational integrators, and nonlinear PDEs by Marsden, Patrick, Shkoller.
- Discrete Geometric Mechanics for Variational Time Integrators by Stern, Desburn
- Variational Integrators, a post by Rod Carvalho with relevant references
- Jerrold Marsden on Discrete Mechanics and Optimal Control
- Differential Geometry in Computational Electromagnetics, PhD thesis by Eric Alan Forgy