Modern scientific computing increasingly demands "structure preserving" numerical methods that faithfully capture underlying structures such as conservation laws, compatibility conditions, and geometric invariants. In many fundamental settings, from elasticity to relativity, these structures are inherently geometric, built using metric, curvature, covariant derivatives, and related quantities. Encoding geometry into reliable predictive computational frameworks has emerged as an interesting challenge at the interface of analysis, geometry, and numerics.

In this talk, I present one of my research lines aimed at incorporating geometry into finite element methods. Riemannian metrics can be approximated by piecewise-defined fields called Regge metrics. I describe a generalization of curvature for such nonsmooth fields that enables, for the first time, high-order finite element approximations of curvature with provable convergence rates in arbitrary dimensions. The heart of the matter actually goes beyond any utilitarian considerations of numerics: our construction give a generalized notion of Riemann curvature on isometrically glued manifolds whose intrinsic metric is not necessarily flat. I also discuss ongoing work on distributional generalizations of extrinsic curvature, emphasizing how key geometric insights guide the analysis.

More broadly, this work gives a way forward to systematically develop compatible discretizations of geometric structures—metrics, connections, curvature, and related tensor fields—within finite element methods. I conclude by outlining longer-term goals such as rigorous finite element methods for geometric flows and partial differential equations arising in continuum mechanics and numerical relativity.