RAFFINE

The RAFFINE project involves the development of a posteriori estimators and adaptive methods for integral equations in the field of simulation of acoustic, electromagnetic and elastic waves. Integral methods are one of the major tools for the numerical simulation of wave propagation and scattering. Indeed, these methods have undergone major development, from both the theoretical and the numerical viewpoints, over the last two decades, particularly with the advent of the Fast Multipole Method (FMM). Paradoxically, very few practical tools exist that allow to control the quality of the discrete solution and the impact of the discretization error on quantities of interest (such as the Radar Cross Section or the scattering matrix). Our project aims to fill this gap. For domain finite element methods, error estimation and control is classically performed using a posteriori estimators of various kinds, about which an abundant literature is available, and tools for adaptivity having proven efficiency. For integral equations, the theoretical literature is much scarcer, and attempts to implement existing estimators in commercial or industrial codes are very rare. We first propose to explore several theoretical ways (residual-based or "goal oriented" estimators) that are now classical for domain finite element methods, assessing and adapting them to the specific difficulties raised by integral equations (nonlocal integral operators, singularity of the kernel). These estimators will be used to define and implement (i) meshing or remeshing strategies, and (ii) stopping criteria for iterative linear solvers. We will then propose efficient algorithms that are implementable for large-scale boundary element models, especially by exploiting parallelization strategies and acceleration by the FMM. Finally, the tools developed will be assessed and validated on canonical cases, illustrating the typical difficulties encountered, and then on particularly demanding industrial cases.