Abstract: The dual argument is well known for recoving the optimal L₂-error of the finite element method in elliptic context. This argument, however, will lose efficacy in hyperbolic case. An expansion argument and an approximation argument are presented in this paper to recover the optimal L₂-error of finite element methods for hyperbolic problems. In particular, a second order error estimate in L₂-norm for the standard linear finite element method of hyperbolic problems is obtained if the exact solution is smooth and the finite element mesh is almost uniform, and some superconvergence estimates are also established for leas smooth solution.