Numerical analysis for the wave equation with locally nonlinear distributed damping

In this paper, we present spectral methods in order to solve wave equation subject to a locally distributed nonlinear damping. Thanks to the efficiency and the accuracy of spectral method, we can check that discrete energy decreases to zero as time goes to infinity, uniformly with respect to the mesh size when the damping is supported in a suitable subset of the domain of consideration. We prove the convergence of the full Fourier–Galerkin discretization. Thus, we apply our schemes to illustrate the uniform discrete energy decay rates of the solution for a wide range of damping functions.