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

TL;DR

This paper introduces a generalized finite element method tailored for the strongly damped wave equation with highly variable coefficients, effectively handling damping and wave speed variations with proven optimal convergence.

Contribution

The paper develops a novel localized orthogonal decomposition-based finite element method that automatically adapts to variations in damping and wave speed, ensuring optimal convergence.

Findings

Convergence proven in $L_2(H^1)$-norm, independent of coefficient derivatives

Numerical examples confirm theoretical convergence rates

Method effectively handles highly varying coefficients in damped wave equations

Abstract

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 introduced 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 in $L_2(H^1)$-norm, independent of the derivatives of the coefficients. We present numerical examples that confirm the theoretical findings.