Section: New Results

Boundary integral formulations of wave scattering

We have continued to develop and further analyze new boundary integral formulation for wave scattering by complex objects.

In [13] we considered acoustic scattering of time-harmonic waves at objects composed of several homogeneous parts. Some of those may be impenetrable, giving rise to Dirichlet boundary conditions on their surfaces. We started from the second-kind boundary integral approach of [X. Claeys, and R. Hiptmair, and E. Spindler. A second-kind Galerkin boundary element method for scattering atcomposite objects. BIT Numerical Mathematics, 55(1):33-57, 2015] and extended it to this new setting. Based on so-called global multi-potentials, we derived variational second-kind boundary integral equations posed in $L^2\left(\Sigma \right)$, where $\Sigma$ denotes the union of material interfaces. To suppress spurious resonances, we introduced a combined-field version (CFIE) of our new method. We conducted thorough numerical tests that highlighted the low and mesh-independent condition numbers of Galerkin matrices obtained with discontinuous piecewise polynomial boundary element spaces. They also confirmed competitive accuracy of the numerical solution in comparison with the widely used first-kind single-trace approach.

We spent much effort investigating the potentialities of multi-trace formulations in terms of domain decomposition. We considered multi-trace formulations in this perspective. Indeed Multi-Trace Formulations are based on a decomposition of the problem domain into subdomains, and thus domain decomposition solvers are of interest. The fully rigorous mathematical MTF can however be daunting for the non-specialist. In [12] , we introduced MTFs on simple model problems using concepts familiar to researchers in domain decomposition. This allowed us to get a new understanding of MTFs and a natural block Jacobi iteration, for which we determined optimal relaxation parameters. We then showed how iterative multitrace formulation solvers are related to a well known domain decomposition method called optimal Schwarz method: a method which used Dirichlet to Neumann maps in the transmission condition. We finally showed that the insight gained from the simple model problem leads to remarkable identities for Calderón projectors and related operators, and the convergence results and optimal choice of the relaxation parameter we obtained is independent of the geometry, the space dimension of the problem, and the precise form of the spatial elliptic operator, like for optimal Schwarz methods. We confirmed this analysis with numerical experiments.

This work was extended in [10] . Considering pure transmission scattering problems in piecewise constant media, we derived an exact analytic formula for the spectrum of the corresponding local multi-trace boundary integral operators in the case where the geometrical configuration does not involve any junction point and all wave numbers equal. We deduced from this the essential spectrum in the case where wave numbers vary. Numerical evidences of these theoretical results were obtained in 2D.

Finally, in connection with boundary integral formulations, we extended the past work of $[$X. Claeys and R. Hiptmair, Integral equations on multi-screens. Integral Equations and Operator Theory, 77(2):167–197, 2013$]$ where we had developed a framework for the analysis of boundary integral equations for acoustic scattering at so-called multi-screens, which are arbitrary arrangements of thin panels made of impenetrable material. In [3] we extended these considerations to boundary integral equations for electromagnetic scattering.

Viewing tangential multi-traces of vector fields from the perspective of quotient spaces we introduced the notion of single-traces and spaces of jumps. We also derived representation formulas and established key properties of the involved potentials and related boundary operators. Their coercivity were proved using a splitting of jump fields. Another new aspect emerged in the form of surface differential operators linking various trace spaces.