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

TL;DR

The paper analyzes uniform exponential decay and optimal error control for semidiscrete (in space) finite element approximations of the linear weakly damped wave equation $u''(t) + \alpha u'(t) + Au(t) = 0$ on convex domains, with $A$ a self-adjoint elliptic operator. Using energy methods, it proves that the semidiscrete solution preserves the continuous decay rate within a specified range $0<\delta \le \tfrac13 \min(\alpha, \lambda_1/(2\alpha))$, and establishes optimal error estimates under minimal smoothness of the initial data, including $L^{\infty}$-type bounds in 2D and higher-order energies. The analysis is extended to generalized settings: nonhomogeneous forcing, space-dependent damping, viscous damping with compensator, and weakly damped beam equations, showing that decay rates can be improved under stabilization. Numerical experiments corroborate the theoretical results and illustrate the practical stability and convergence properties of the semidiscrete and fully discrete schemes. The work provides rigorous guarantees for long-time behavior and accuracy of FE discretizations of damped wave systems, with implications for simulations in acoustics, elasticity, and related fields.

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.

Figures (8)

• Figure 1: Example 1: The decay estimates in $L^{\infty}, \, L^2$ and $H^{1}$-norms.
• Figure 2: Example 2: The decay estimates in $L^{\infty}, \, L^2$ and $H^{1}$-norms for $\alpha_0 = 1$ and $\gamma = \frac{1}{2}$.
• Figure 3: Example 3: The decay estimates in $L^{\infty}, \, L^2$ and $H^{1}$-norms for $\alpha = 4$.
• Figure 4: Example 4: The decay estimates in $L^{\infty}, \, L^2$ and $H^{1}$-norms for $\alpha = 1$.
• Figure 5: Example 4: The decay estimates in $L^2$ and $L^{\infty}$-norms for different $\alpha$ values.

Theorems & Definitions (19)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Remark 2.1
• Remark 2.2
• Theorem 3.1
• Theorem 3.2
• Remark 3.1
• Theorem 3.3