Error analysis with exponential decay estimates for a fully discrete approximation of a class of strongly damped wave equations

TL;DR

This paper analyzes the exponential decay and error estimates of a fully discrete finite element and finite difference scheme for strongly damped wave equations, ensuring stability and decay properties are preserved in discretization.

Contribution

It introduces a novel approach for deriving exponential decay estimates for damped wave equations and provides optimal error bounds for a fully discrete scheme that maintains decay behavior.

Findings

Exponential decay rates are explicitly derived for different damping parameters.

Optimal error estimates are established that preserve decay properties.

Numerical experiments confirm theoretical decay rates and stability.

Abstract

This paper deals with the asymptotic behavior and FEM error analysis of a class of strongly damped wave equations using a semidiscrete finite element method in spatial directions combined with a finite difference scheme in the time variable. For the continuous problem under weakly and strongly damping parameters $\alpha$ and $\beta,$ respectively, a novel approach usually used for linear parabolic problems is employed to derive an exponential decay property with explicit rates, which depend on model parameters and the principal eigenvalue of the associated linear elliptic operator for the different cases of parameters such as $(i) \;\alpha, \beta >0$, $ (ii)\; \alpha>0, \beta \geq 0$ and $(iii)\;\alpha \geq 0, \beta >0$. Subsequently, for a semi-discrete finite element scheme keeping the temporal variable continuous, optimal error estimates are derived that preserve exponential decay behavior. Some generalizations that include forcing terms and spatially as well as time-varying damping parameters are discussed. Moreover, an abstract discrete problem is discussed, and as a consequence, uniform decay estimates for finite difference as well as spectral approximations to the damped system are briefly indicated. A complete discrete scheme is developed and analyzed after applying a finite difference scheme in time, which again preserves the exponential decay property. The given proofs involve several energies with energy-based techniques to derive the consistency between continuous and discrete decay rates, in which the constants involved do not blow up as $\alpha\to 0$ and $\beta\to 0$. Finally, several numerical experiments are conducted whose results support the theoretical findings, illustrate uniform decay rates, and explore the effects of parameters on stability and accuracy.