A conservative fully discrete finite element scheme for the nonlinear Klein–Gordon equation

This article proposes a family of the Petrov–Galerkin–FEM methods that can be used to solve the nonlinear Klein–Gordon equation. The discrete schemes were formulated based on the solution of the problem and its time derivative. They ensure that the total energy is conserved at a discrete level. The simplest two-layer scheme was studied numerically. Based on the solution of the test problems with smooth solutions, it was shown that the scheme can determine the solution of the problem, as well as its time derivative with an error of the order of $O(h^2+\tau^2)$ in the continuous $L_2$ norm, where $\tau$ and $ h$ characterize the grid steps in time and space, respectively.