Upper semicontinuity of pullback attractors for a nonautonomous damped wave equation

In this paper, we study the local uniformly upper semicontinuity of pullback attractors for a strongly damped wave equation. In particular, under some proper assumptions, we prove that the pullback attractor $\{A_{\varepsilon }(t)\}_{t\in \mathbb{R}}$ { A ε ( t ) } t ∈ R of Eq. (1.1) with $\varepsilon \in [0,1]$ ε ∈ [ 0 , 1 ] satisfies $\lim_{\varepsilon \to \varepsilon _{0}}\sup_{t\in [a,b]} \operatorname{dist}_{H_{0}^{1}\times L^{2}}(A_{\varepsilon }(t),A_{ \varepsilon _{0}}(t))=0$ lim ε → ε 0 sup t ∈ [ a , b ] dist H 0 1 × L 2 ( A ε ( t ) , A ε 0 ( t ) ) = 0 for any $[a,b]\subset \mathbb{R}$ [ a , b ] ⊂ R and $\varepsilon _{0}\in [0,1]$ ε 0 ∈ [ 0 , 1 ] .

A generalized finite element method for the strongly damped wave equation with rapidly varying data

We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition, and is designed to handle independent variations in both the damping and the wave propagation speed respectively. The method does so by automatically correcting for the damping in the transient phase and for the propagation speed in the steady state phase. Convergence of optimal order is proven, independent of the derivatives of the coefficients. We present numerical examples that confirm the theoretical findings.