Abstract

We are interested in the numerical solution of time-harmonic scattering by complex geometries. We will first consider so-called multi-screens, which are geometries composed of panels meeting at junction lines. This is modelled via first kind integral equations using the framework proposed by Claeys and Hiptmair. The key realisation is that solutions of the related boundary integral equations belong to jump spaces, that can be represented as the quotient-space of a multi-trace space and a single trace space. As shown in previous work, the corresponding Galerkin discretization via quotient-space boundary element methods is up to the task. However, it does not address the ill-conditioning of the arising Galerkin matrices and the performance of iterative solvers deteriorates significantly when increasing the mesh refinement. As a remedy, we introduce a Calder\’on-type preconditioner and discuss two possible multi-trace discretizations. First, we work with the full multi-trace discrete space, which contains many more degrees of freedom (DoFs) than strictly required. Then, we propose a representation of the quotient-space that reduces significantly the number of degrees of freedom while still allowing for efficient Calder\’on preconditioning. For this, we exploit the fact that the solution to the scattering problem is determined only up to a function in the single trace space. This implies that if we modify the single trace subspace of the multi-trace discrete space, the solution, as an element of the quotient-space, is unaffected.    Finally, we will also consider time-harmonic scattering by composite structures including multiple domains and metallic coatings that can contain junctions. This is achieved by combining the global multi-trace method with the quotient space discretisation of the multi-screen boundary integral equation.    This is joint work with Kristof Cools.