Mathematical models are a foundational part of how mathematicians, scientists and engineers describe and study the behaviour of systems. These models typically include and interact with data: Experimental measurements are necessary inputs to a modelled physical system to produce predictive data outputs. Within a model, data in one part of the modelled system may have a relationship with another part of the modelled system. One may even wish to combine models to study a larger system: these typically interact by producing and consuming each other's data.

Finite Element Methods (FEMs) are a popular method of computing mathematical models which involve systems of equations defined on some domain. In this work, new abstractions are introduced for (a) representing arbitrary point data as finite element functions on disconnected meshes of vertices, and for (b) interacting with that data via interpolation operations which represent point evaluations. These abstractions are consistent with the finite element method paradigm of finite element functions on meshes. The interpolation operations are differentiable, allowing inverse problems to be solved involving point data.

The abstractions are sufficiently general that they allow one to reason about more generic data, both within and external to a given finite element model. The differentiable point evaluation operation allows differentiable interpolation operations between finite element functions on arbitrary coincident meshes.

Whilst these abstractions are generally applicable across many finite element method software packages, they are demonstrated here with an implementation in Firedrake. Firedrake is an MPI-parallelised code generation system which solves variational problems using the finite element method. The implementation is parallel safe and maintains the high level of abstraction described. Demonstrated applications include point-data assimilation, point forcing, model diagnostics gathering, and model coupling. Improvements to the abstractions Firedrake uses for specifying variational problems and reasoning about finite elements are also implemented.