The paper presents a parallel approach for the efficient solution of a one-dimensional and a two-dimensional problem by parallel finite element method. These problems are case studies. The non-overlapping domain decomposition method has been used to cut the problem into subregions or also called sub-domains, and it reduces the large mass matrix into smaller parts. The independent sub-domains, and the assembling of these equation systems can be handled by the independent processors of a supercomputer, i.e. in a parallel way. The execution time and speedup of parallel finite element method have been compared to the serial one.