The paper presents a parallel approach for the efficient solution of a simple 1D Laplace-Poisson equation problem by parallel finite element method. This problem is a case study. The non-overlapping domain decomposition method has been used to cut the problem into sub-regions or also called sub-domains, and it reduces the large mass matrix into smaller parts. The independent subdomains, and the assembling of these equation systems can be handled by the independent processors of a supercomputer, i.e. in a parallel way. The results of parallel finite element method have been compared to the results of serial finite element method as well as the analytical results.