Domain truncation methods for the wave equation in a homogenization limit. Issue 12 (13th August 2022)

• Record Type: Journal Article
• Authors: Schäffner, Mathias; Schweizer, Ben; Tjandrawidjaja, Yohanes
• Is Part Of: Applicable analysis. Volume 101:Issue 12(2022)
• Page Start: 4149
• Page End: 4170
• Publication Date: 2022-08-13
• Subjects: Wave equation -- homogenization -- domain truncation; 35L05 -- 35B27; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2022.2054416
• Languages: English
• ISSNs: 0003-6811

Abstract

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.