Nonlinear wave equation with Dirichlet and Acoustic boundary conditions: theoretical analysis and numerical simulation

We investigate some theoretical and numerical aspects of a nonlinear wave equation with variable coefficient and Dirichlet and Acoustic boundary conditions. The existence and uniqueness of the solution are obtained applying the Faedo-Galerkin method with some compactness results and energy method. In addition, we prove the uniform stability of the energy. For numerical simulation, firstly we use the Crank–Nicolson Galerkin method, in which it consists of applying the finite element method in the spatial variable and the Crank–Nicolson method over time. Subsequently, in the resulting nonlinear algebraic system, for each discrete time, we apply the Newton’s method without losing the convergence order. Moreover, are presented figures of the numerical solutions for the two-dimensional case, tables with error and convergence order and the numerical energy decay. These results justify the consistency between the theoretical and numerical results.