Uniform boundary stabilization of a high-order finite element space discretization of the 1-d wave equation

The aim of this work is to construct and analyze a discretization process that preserves exponential stability at the discrete level for a wave propagation problem with boundary damping when a high-order spectral finite element approximation is used. The main idea is to add a stabilizing term to the wave equation that damps the spurious oscillatory components of the solutions. This term is based on a discrete multiplier analysis that ensures the exponential stability of the semi-discrete problem at any order without compromising the rate of convergence of the discretization strategy. Possible applications of this work concern the solution of boundary control problems or data assimilation problems whose efficiency is based on an exponential stability property of an underlying discrete error system.