The Black-Scholes model is a crucial instrument in financial mathematics for calculating the numerical value of financial options. This article presents a computational method for numerically solving Black-Scholes PDEs governing European options. First, we employ a uniform mesh to discretize the time derivative, and then, we use the backward-Euler method to approximate it. We apply a piecewise uniform Shishkin mesh for spatial domain discretization and the streamline-diffusion finite element method to compute the spatial derivative. We have performed a series of numerical computations to validate the theoretical convergence rates.