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

TL;DR

This paper analyzes a class of strongly damped wave equations $u'' + \beta Au' + \alpha u' + Au = 0$ with $A=-\Delta$ and Dirichlet boundary conditions, deriving explicit exponential decay rates that depend on the damping parameters $\alpha,\beta$ and the principal eigenvalue $\lambda_1$. It develops energy-based proofs for the continuous problem and for semidiscrete and completely discrete finite element/time-stepping schemes, showing that discretizations can preserve the continuous decay uniformly in the discretization parameters. It provides optimal error estimates in $L^\infty(L^2)$ and $L^\infty(H^1)$ norms under minimal initial regularity, along with higher-order results and $L^\infty$ bounds in two dimensions, and extends to forcing terms and time/space-varying damping. Numerical experiments corroborate the theory, demonstrating uniform decay, stability, and convergence across parameter regimes, and illustrating practical guidance for choosing damping properties in applications.

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 $α$ and $β,$ 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 $(i) \;α, β>0$, $ (ii)\; α>0, β\geq 0$ and $(iii)\;α\geq 0, β>0$. 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 $α\to 0$ and $β\to 0$. 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.

Figures (8)

• Figure 1: Example \ref{['Exm1']}: The plots of discrete energy $E_h(u_h)$ and continuous energy $E(u)$.
• Figure 2: Example \ref{['Exm1']}: Discrete energy decay profiles and estimate of decay rate.
• Figure 3: Example \ref{['Exm2']}: The plots of discrete energy $E_h(u_h)$ and continuous energy $E(u)$.
• Figure 4: Example \ref{['Exm2']}: Discrete energy decay profiles and estimate of decay rate.
• Figure 5: Example \ref{['Exm3']}$(i)$: The plots of discrete energy $E_h(u_h)$ and continuous energy $E(u)$.

Theorems & Definitions (46)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• Remark 2.1
• Remark 2.2
• Theorem 3.1