Dynamics in Hamiltonian Lattice Gauge Theory: Approaching the Continuum Limit with Partitionings of SU$(2)$

In this paper, we investigate a digitised SU$(2)$ lattice gauge theory in the Hamiltonian formalism. We use partitionings to digitise the gauge degrees of freedom and show how to define a penalty term based on finite element methods to project onto physical states of the system. Moreover, we show for a single plaquette system that in this framework the limit $g\to 0$ can be approached at constant cost.