Dimensional reduction for Helmholtz's equation on a bounded domain

Abstract

The dimensional reduction method for solving boundary value problems of Helmholtz's equation in domain $\Omega^d:=\Omega\times (-d,d)\subset {\Bbb R}^{n+1}$ by replacing them with systems of equations in $n-$ dimensional space are investigated. It is proved that the existence and uniqueness for the exact solution $u$ and the dimensionally reduced solution $u_N$ of the boundary value problem if the input data on the faces are in some class of functions. In addition, the difference between $u$ and $u_N$ in $H^1(\Omega^d)$ is estimated as $d$ and $N$ are fixed. Finally, some numerical experiments in a domain $\Omega=(0,1)\times (0,1)$ are given in order to compare theretical results.