Trefftz method for electromagnetic wave simulation in three dimensions

Abstract

International audienceThe simulation of time-harmonic electromagnetic waves requires a matrix inversion whose cost, especially in three-dimensional cases, increases quickly with the size of the computational domain. This is a tangible issue regarding memory consumption when the size of the domain comes of a few dozen wavelengths in each direction. The so-called pollution effect tends to accentuate this problem. This phenomena forces to increase the number of discretization points per wavelength to ensure the accuracy of the numerical solution when the size of the domain increases.A conventional idea consists in reducing computing costs by performing domain decomposition. This method requires the resolution of smaller auxiliary problems on each subdomain. These subdomains are then coupled thanks to Robin fluxes which ensure the convergence of the method. These methods are efficient but not flexible enough to be integrated easily in industrial codes.Recently, lots of authors have been studying Trefftz methods. Such methods offer a good flexibility of the mesh regarding both forms and sizes of cells. Therefore, they can deal with complex geometrical constraints of industrial environment. Trefftz methods consist in using a discontinuous Galerkin method whose basis functions are defined as  local solutions of the studied equation. They can be given either analytically by a sum of plane waves or numerically by an auxiliary solver. These basis functions are specific to the considered physical problem and thus reduce numerical dispersion phenomena. Trefftz methods are also particularly adapted to domain decomposition methods. It is then possible to come up with an iterative Trefftz solver.In this presentation, we will show different Trefftz methods for solving time-harmonic Maxwell problem in three dimensions. We will deal with the case of an auxiliary analytical solver using plane waves, and with the case of an auxiliary numerical solver using high order Nédélec finite elements. A comparison between different formulations will be given. We will also pay particular attention to the accuracy of the method and to the memory necessary for the resolution. Finally, reasons why these numerical methods are adapted to modern architectures will be brought to the fore