Domain truncation methods for the wave equation in a homogenization limit. Issue 12 (13th August 2022) - Schäffner, Mathias; Schweizer, Ben; Tjandrawidjaja, Yohanes
Abstract : We consider the wave equation ∂ t 2 v ϵ − ∇ ⋅ ( a ϵ ∇ ) v ϵ = f on an unbounded domain Ω ∞ for highly oscillatory coefficients a ϵ with the scaling a ϵ ( x ) = a ( x / ϵ ) . We consider settings in which the homogenization process for this equation is well understood, which means that v ϵ → v ¯ holds for the solution v ¯ of the homogenized problem ∂ t 2 v ¯ − ∇ ⋅ ( a ∗ ∇ ) v ¯ = f . In this context, domain truncation methods are studied. The goal is to calculate an approximate solution u ϵ 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 u ϵ − v ϵ . We present different methods and corresponding O ( ϵ ) error estimates.