Exponential decay of the discrete energy for the wave-wave coupled system

TL;DR

This paper analyzes the decay of energy in a dissipative coupled wave system using finite element and Euler methods, proving exponential decay of the discrete energy for the first time.

Contribution

It introduces a novel proof of exponential energy decay for a fully discrete coupled wave system using numerical methods.

Findings

Proves linear convergence of the numerical scheme.

Establishes exponential decay of the discrete energy.

Provides a priori error estimates under regularity assumptions.

Abstract

In this article, a numerical analysis of the asymptotic behavior of the discrete energy associated to a dissipative coupled wave system is conducted. The numerical approximation of the system is constructed using the P1 finite element method for spatial discretization, combined with the implicit Euler scheme for time integration. An a priori error analysis is established, showing that, under extra regularity assumptions on the continuous solution, the numerical scheme exhibits linear convergence. Then, for the first time in the literature, the exponential decay of the fully discrete energy is shown using the energy method.