A category theoretical interpretation of discretization in Galerkin finite element method (Q2205603)

In category theory, the notion ``discrete'' has a precise meaning. The authors introduce the notion of discretisation (``the unique dagger mono representative of a subobject of \(\mathcal{H}\), discrete with respect to \(U: \mathbf{Hilb} \to \mathbf{Vec}\)''). They note that discretisation, as performed by numerical analysts in the Galerkin method, fits this abstract description; this is a reformulation of the elementary statement that a Hilbert space \(\mathcal{H}\) is finite-dimensional if and only if every linear mapping defined on \(\mathcal{H}\) is continuous. They also define ``parallel decomposability'' of discretisations and perform a case study in magnetostatics. This approach does not seem to have a bearing on the rate of convergence for the Galerkin method.