Dissertation (MSc Mathematics)Over the previous years finite element method (FEM) has become a powerfully tool to approximate solution of differential equations and prove their existence. The purpose of this research is to introduce and describe a number of the finite element method (FEM) technique applied to problems for partial differential equations (PDEs) with special attentions to the hyperbolic problems in case of wave and damped wave equations. Another aim is to study the one boundary value problem (BVP) for the wave equation and apply damping control multi-step methods integrated into the FEM such as the Newmark method, Backward difference method (BDF) and Hilber-Hughes-Taylor Method (HHT). The ordinary differential equation (ODE) system obtained after applying FEM are then solved by these multi-step methods, where by the BDF-Method and the HHT-Method are second order precision, unconditionally stable and able to dissipate high-modes for some values of the parameters.