Abstract

This paper presents the numerical analysis of a one-dimensional vibrating string coupled with a dynamic boundary control system modeling acoustic interactions. The system, governed by a wave equation and an ordinary differential equation at the boundary, exhibits limited regularity due to the presence of unbounded trace terms. To overcome this, a weak formulation is adopted and discretized using the finite element method in space and the Crank-Nicolson scheme in time. A key feature of the analysis is a regularity transfer mechanism that allows us to derive meaningful error estimates despite the lack of classical smoothness. Numerical simulations confirm the polynomial decay of energy and emphasize the critical role of the coupling parameter \( C \) in accelerating dissipation. The results validate the efficiency of the control and the reliability of the numerical scheme.

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