Single Scattering Preconditioner Applied to Boundary Integral Equations

Paris, France

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 $\Omega_p$. The boundary condition is here set to Dirichlet but another condition can be imposed.

It is well known that this problem can be rewritten equivalently under the form of a system of boundary integral equations (BIEs) with the densities $\rho$ and $\lambda$ as unknowns. If $\mathcal{L}$ and $\mathcal{M}$ represent respectively the volume single- and double-layer integral operators, then $$ u(x) = \mathcal{L}\rho(x) + \mathcal{M}\lambda(x), \qquad \forall x \not\in\cup_{p=1}^M\overline{\Omega_p}. $$ Following 1, the BIE can be classified as direct or indirect, depending on whether or not the unknown dentities $\rho$ and $\lambda$ are the Cauchy data. Direct BIE here also refers to the null-field method.

In multiple scattering context, a natural preconditioner is the one representing single scattering effects. For M obstacles and given the matrix of a discretized integral equation, this preconditioner is composed by the M blocks located on the diagonal of this matrix. Each block represents the scattering problem by one obstacle. This geometric preconditioner is called single scattering preconditioner.

This talk focuses on the effects of this preconditioning on boundary integral equations. The main result is that, after being preconditioned by their single scattering preconditioner, every direct integral equations become exactly the same. This does not depend on the geometry of the obstacles and can moreover be extended in a different form for indirect integral equations such as the one of Brakhage-Werner. These properties imply in particular that the convergence rate of a Krylov subspaces solver will be exactly the same for every preconditioned integral equations.

  1. A. Bendali and M. Fares, Computational Methods for Acoustics Problems, chapter Boundary Integral Equations Methods in Acoustics, Saxe-Coburg Publications, 2007 ^
Bertrand Thierry

Permanent researcher of the CNRS, working in numerical simulation for wave propagation.