Trefftz methods for solving large-scale time-harmonic wave problems
Résumé
Wave propagation is a complex physical phenomenon that makes the invisible visible by solving an inverse problem. The underlying mathematical model can be formulated in either the time or frequency regime, each with its own advantages and disadvantages. Here, we prefer the frequency domain, which makes it easier to take into account physical parameters such as attenuation. In this case, direct problem solving, crucial in the inversion algorithm, is very costly and the size of the system to be solved quickly reaches the limits of direct linear solvers. Here, we propose a numerical method that relaxes memory constraints by adopting an iterative approach. To this end, we construct an iterative method that belongs to the class of Discontinuous Galerkin Trefftz methods. By reducing calculations to the level of the mesh skeleton, these are known to use less memory than conventional finite element methods. The method is explained in detail in the 1D case, describing its main features, which are a legacy of Trefftz's idea of using approximation spaces composed of special functions, in this case plane waves. This leads to a wellposed discrete problem that can be solved by an iterative block Jacobi method. The method's performance is illustrated in 1D and 3D.
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