  This paper presents and analyzes a parallelizable iterative procedure based
on domain decomposition for primal-dual weak Galerkin (PDWG) finite element
methods applied to the Poisson equation. The existence and uniqueness of the
PDWG solution are established. Optimal order of error estimates are derived in
both a discrete norm and the $L^2$ norm. The convergence analysis is conducted
for domain decompositions into individual elements associated with the PDWG
methods, which can be extended to larger subdomains without any difficulty.
