Domain truncation methods for the wave equation in a homogenization limit
Mathias Schäffner, Ben Schweizer, Yohanes Tjandrawidjaja · Applicable Analysis · 2022
We consider the wave equation ∂t2vϵ−∇⋅(aϵ∇)vϵ=f on an unbounded domain Ω∞ for highly oscillatory coefficients aϵ with the scaling aϵ(x)=a(x/ϵ). We consider settings in which the homogenization process for this equation is well understood, which means that vϵ→v¯ holds for the solution v¯ of the homogenized problem ∂t2v¯−∇⋅(a∗∇)v¯=f. In this context, domain truncation methods are studied. The goal is to calculate an approximate solution uϵ on a subdomain, say Ω−⊂Ω∞. We are ready to solve the ε-problem on Ω−, but we want to solve only homogenized problems on the unbounded domains Ω∞ or Ω∞∖Ω¯−. The main task is to define transmission conditions at the interface to have small differences uϵ−vϵ. We present different methods and corresponding O(ϵ) error estimates.