The Boltzmann equation (BE) for rarefied gas dynamics presents significant computational challenges due to its high dimensionality in phase space, requiring substantially more memory and computational resources than equivalent Navier-Stokes simulations. A domain decomposition approach is presented to address these challenges by distributing the computational load across high-performance computing infrastructure. A finite element formulation with discontinuous Galerkin (dG) spatial discretization and the method of moments for velocity space is employed. The spatial domain is partitioned into subdomains that exchange flux information at interfaces during each timestep, with inflow to one subdomain corresponding to outflow from its neighbour. This framework enables several key advantages: reduced system matrix sizes and improved memory efficiency. Entropy-based closure relations are incorporated within the method of moments approach to ensure physically consistent entropy dissipation. Anderson acceleration is employed to minimize subiteration overhead within each timestep. The closure also necessitates assigning a prior distribution, which, along with the velocity discretization, can be varied between different subdomains based on the local degree of rarefaction. Therefore, finer discretizations can be chosen (with better prior guesses) only where highly rarefied flows are expected, which results in further reduction of computational overhead. As an example, fluid dynamics within a channel connected to reservoirs with different pressures is analysed, with the decomposed domain solutions demonstrating excellent agreement with monolithic solutions across a range of Knudsen numbers.