Domain truncation methods for the wave equation in a homogenization limit

Abstract

We consider the wave equation ${\mathrm{\partial }}_t^2{v}^{ϵ}-\mathrm{\nabla }\cdot \left(a_{ϵ}\mathrm{\nabla }\right)v^{ϵ}=f$ on an unbounded domain ${\mathrm{\Omega }}_{\mathrm{\infty }}$ for highly oscillatory coefficients $a_{ϵ}$ with the scaling $a_{ϵ}\left(x\right)=a\left(x/ϵ\right)$. We consider settings in which the homogenization process for this equation is well understood, which means that $v^{ϵ}\to \overline{v}$ holds for the solution $\overline{v}$ of the homogenized problem ${\mathrm{\partial }}_t^2\overline{v}-\mathrm{\nabla }\cdot \left(a_{\ast }\mathrm{\nabla }\right)\overline{v}=f$. In this context, domain truncation methods are studied. The goal is to calculate an approximate solution $u^{ϵ}$ on a subdomain, say ${\mathrm{\Omega }}_{-}\subset {\mathrm{\Omega }}_{\mathrm{\infty }}$. We are ready to solve the ε-problem on ${\mathrm{\Omega }}_{-}$, but we want to solve only homogenized problems on the unbounded domains ${\mathrm{\Omega }}_{\mathrm{\infty }}$ or ${\mathrm{\Omega }}_{\mathrm{\infty }}\setminus {\overline{\mathrm{\Omega }}}_{-}$. 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 $O\left(ϵ\right)$ error estimates.

Keywords:

• Wave equation
• homogenization
• domain truncation

MSC:

• 35L05
• 35B27

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant SCHW 639/11-1, ‘Strahlungsbedingungen für Wellen in periodischen und stochastischen Medien’.