This thesis concerns the implementation of parallel Schwarz domain decomposition using node-level parallelism, focusing on the parallel Schwarz method in comparison with the Jacobi iterative method. The study goes into the complexities of domain decomposition methods for solving partial differential equations, which are essential in fields such as fluid dynamics, solid mechanics, quantum mechanics, and financial mathematics. The research examines the convergence and performance of these methods within a parallel computing framework. A large portion of the work involves the comparison of varying configurations of block sizes and overlaps within the use of the block Jacobi iterative method, used for the implementation of the parallel Schwarz method. Numerical experiments are conducted for a stationary heat problem on a 2-dimensional grid with 256 points in each direction. The results show optimal performance for small block sizes, attributed to the use of a dense solver for the subdomains. Larger blocks and larger overlaps show superior convergence properties, up to the limit of an overlap of half the block size. The efficiency of the parallel Schwarz method remains high for an increasing number of threads unlike the standard Jacobi iteration, showing it is better suitable to a parallel environment.