Uniform indirect boundary observability for a spatial discretization of strongly coupled wave equations

This paper investigates the indirect boundary observability properties of one-dimensional strongly coupled wave equations in an approximated setting. Classical numerical discretization methods, such as finite differences and finite elements, typically fail to maintain uniform observability inequalities when applied to wave systems. This failure is primarily attributed to the emergence of high-frequency numerical solutions. The present work demonstrates a different approach through the implementation of these discretization schemes on a carefully designed non-uniform mesh. This study successfully establishes uniform observability inequalities for the coupled system. This methodology effectively recovers the system’s total energy through boundary observations, overcoming the well-documented limitations of traditional numerical approaches in wave equation systems.