Approximation of periodic solutions for a dissipative hyperbolic equation

Abstract

This paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force \(f\). These approximations are obtained by combining a fixed point algorithm with the Galerkin method. It is known that the energy of the usual discrete models does not decay uniformly with respect to the mesh size. Our aim is to analyze this phenomenon’s consequences on the convergence of the approximation method and its error estimates. We prove that, under appropriate regularity assumptions on \(f\), the approximation method is always convergent. However, our error estimates show that the convergence’s properties are improved if a numerically vanishing viscosity is added to the system. The same is true if the nonhomogeneous term \(f\) is monochromatic. To illustrate our theoretical results we present several numerical simulations with finite element approximations of the wave equation in one or two dimensional domains and with different forcing terms.