A numerical solution to Klein-Gordon equation with Dirichlet boundary condition

Klein-Gordon equation arises in relativistic quantum mechanics and field theory, so it is of a great importance for the high energy physicists. In this paper, we establish the existence and uniqueness of the solution and in the second part a numerical scheme is developed based on finite element method. For one space dimensional case, a complete numerical algorithm for the numerical solutions using the quadratic interpolation functions is constructed. The one-dimensional model equation is formulated over an arbitrary element, applying the assembly process on the elements of the domain, employing numerical scheme to integrate the nonlinear terms and solving the system of equations numerically. Finally the obtained results of simulation is visualized, which shows the overflow of the solution as expected.