Multiple traces boundary integral formulation for Helmholtz transmission problems

Abstract

We present a novel boundary integral formulation of the Helmholtz transmission problem for bounded composite scatterers (that is, piecewise constant material parameters in "subdomains") that directly lends itself to operator preconditioning via Calderón projectors. The method relies on local traces on subdomains and weak enforcement of transmission conditions. The variational formulation is set in Cartesian products of standard Dirichlet and special Neumann trace spaces for which restriction and extension by zero are well defined. In particular, the Neumann trace spaces over each subdomain boundary are built as piecewise H −1/2 -distributions over each associated interface. Through the use of interior Calderón projectors, the problem is cast in variational Galerkin form with an operator matrix whose diagonal is composed of block boundary integral operators associated with the subdomains. We show existence and uniqueness of solutions based on an extension of Lions' projection lemma for non-closed subspaces. We also investigate asymptotic quasi-optimality of conforming boundary element Galerkin discretization. Numerical experiments in 2-D confirm the efficacy of the method and a performance matching that of another widely used boundary element discretization. They also demonstrate its amenability to different types of preconditioning.

Multitrace Formulations and Dirichlet-Neumann Algorithms

“…The multitrace formulation from [4] states that the pairs (T i u i ) i=1,2 are traces of the solution defined on Ω i if they verify the relations…”

“…Multitrace formulations (MTF) for boundary integral equations (BIE) were developed over the last few years in [4] and [1,2] for the simulation of electromagnetic problems in piecewise constant media, see also [3] for associated boundary integral methods. The MTFs are naturally adapted to the developments of new block preconditioners, as indicated in [5], but very little is known so far about such associated iterative solvers.…”

Electromagnetic scattering at composite objects : a novel multi-trace boundary integral formulation

“…A similar approach has been proposed in [26], but only for the acoustic case. A related technique, the boundary element tearing and interconnecting (BETI) method (a boundary element counterpart of the FETI method) has been developed by Steinbach et al for strongly elliptic problems [28,30,36].…”

Quasi‐local multitrace boundary integral formulations

“…In this section, we will only recall already established results concerning local multitrace formulation, not giving a complete overview, and refer the reader to [13] for detailed proof of these results. A key ingredient of the local multitrace theory is an operator yielding a characterization of transmission conditions of (1).…”