# Asymptotic behaviour of the semidiscrete FE approximations to weakly damped wave equations with minimal smoothness on initial data

## TL;DR

This paper analyzes the asymptotic behavior of semidiscrete finite element approximations to weakly damped wave equations, establishing decay estimates and error bounds with minimal smoothness assumptions, supported by numerical validation.

## Contribution

It provides uniform decay estimates and optimal error bounds for semidiscrete FE methods applied to weakly damped wave equations with minimal initial data smoothness.

## Key findings

- Decay rates match continuous case
- Optimal error estimates under minimal smoothness
- Numerical experiments confirm theoretical results

## Abstract

Exponential decay estimates of a general linear weakly damped wave equation are studied with decay rate lying in a range. Based on the $C^0$-conforming finite element method to discretize spatial variables keeping temporal variable continuous, a semidiscrete system is analysed, and uniform decay estimates are derived with precisely the same decay rate as in the continuous case. Optimal error estimates with minimal smoothness assumptions on the initial data are established, which preserve exponential decay rate, and for a 2D problem, the maximum error bound is also proved. The present analysis is then generalized to include the problems with non-homogeneous forcing function, space-dependent damping, and problems with compensator. It is observed that decay rates are improved with large viscous damping and compensator. Finally, some numerical experiments are performed to validate the theoretical results established in this paper.