Black-Scholes partial differential equation (PDE) is one of the most famous equations in mathematical finance and financial industry. In this study, numerical solution analysis is done for Black-Scholes PDE using finite element method with linear approach and finite difference methods. The numerical solutions are compared with Black-Scholes formula for option pricing. The numerical errors are determined for the finite element and finite difference applications to Black-Scholes PDE. We examine the error behavior and find the source of the corresponding errors under various market situations.