Abstract: The Orr-Sommerfeld equation is solved numerically using a finite element method and the LR matrix eigenvalue algorithm.The results of great accuracy are obtained, by choosing higher order Hermite polynomials to be basis functions and making element distribution according to the physical character of the flow existed in the region.The method is applied to the stability of plane Poiseuille flowi It is found that the critical Reynolds nnmber is 5772.2218 and, it is computed that the neutral curve in the range of Reynolds number from Rc to 1010 farabove than those obtained before