Helmholtz Equation

Single Scattering Preconditioner Applied to Boundary Integral Equations

In a homogeneous medium, when illumated by an incident time-harmonic acoustic wave $u^{inc}$, the $M > 1$ obstacles $\Omega_p, p=1, …,M$, generate a scattered wave $u$ solution of the Helmholtz equation: $$ \left\{ \begin{array}{r c l l} \Delta u + k^2u & = &0 & \mathbb{R}^3\setminus\overline{\cup_{p=1}^M\Omega_p}\\ u & = & -u^{inc} & \cup_{p=1}^M\Gamma_p\\ u & \text{ is } & \text{radiating.} \end{array} \right. $$ The quantity $k$ is the positive wavenumber, the radiating condition stands for the Sommerfeld one and $\Gamma_p$ are the boundaries of $\Omega_p$.