Optimized Schwarz methods for time-harmonic wave problems in resonating cavities

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Numerical simulations of time-harmonic wave problems are nowadays of paramount importance in the development of many key technologies, such as medical imaging (e.g. with applications in rapid stroke detection), particle accelerators (e.g. with applications in cancer treatment), or photonics (e.g. with applications in high-speed telecommunication), just to cite a few.The numerical treatment of large structures (with respect to the considered wavelength) is more and more required, especially in the field of particle accelerators.Indeed, in this case, many cavities are typically organized in long chains, in order to provide the required acceleration to the considered particles, creating thus large structures.In addition to the accelerating mode of the cavities, higher-order parasitic modes must be studied with care, as they might jeopardize the stability of the particle beam, motivating thus the use of numerical computations.However, when considering large structures, the computational effort needed for solving time-harmonic wave problems becomes very large, requiring thus the use of super-computing facilities together with numerical techniques taking advantage of the computational resources.Unfortunately, due to their mathematical nature, only a few techniques are available for efficiently solving large-scale time-harmonic wave problems, and the treatment of large-scale time-harmonic wave problems remains a difficult, if not impossible, task.Obviously, all these techniques have advantages and drawbacks.In the case of optimized Schwarz (OS) methods, which were successfully applied for simulating antenna arrays, photonic waveguides or for reconstructing medical images, it can be shown that their performance drops when treating cavity structures, i.e. structures where the effect of back-propagating waves cannot be neglected.Such cavity structures are found, as already mentioned, in particle accelerators, but also in other key technologies such as lasers or quantum electrodynamic devices.This research proposal aims at developing an accurate and reliable numerical method for solving large-scale time-harmonic wave problems in cavity structures, by alleviating the performance drop exhibited by OS methods when treating back-propagating waves.