This project is concerned with the development of high-performance, efficient and robust techniques for the numerical simulation of complex physical phenomena governed by vector-valued variable-coefficient equations in three dimensions. On the one hand it includes advanced computational methods for problems in strongly inhomogeneous and anisotropic media, with an emphasis on time-harmonic wave propagation problems. On the other hand it includes applications to plasma physics and aeroacoustics. For instance, in the field of waves in plasmas, inhomogeneity comes into play in a non-uniform plasma – when its density is not constant – while anisotropy comes into play in a magnetized plasma – in the presence of an equilibrium magnetic field causing the plasma properties to vary depending on the direction considered. The computational methods under consideration are called quasi-Trefftz methods. Trefftz methods are seeing a renewed interest in the scientific community as various techniques have recently been explored to tame the conditioning issues of wave-based discretizations. Like them, quasiTrefftz methods rely on local problem-dependent basis functions. However, unlike Trefftz basis functions, their basis functions are not exact solutions of the governing equation. They were precisely introduced to leverage the benefits of Trefftz methods while tackle problems governed by variable-coefficient equations, since exact Trefftz basis functions do not exist in general for such problems. This project will develop quasi-Trefftz methods for vector-valued equations in three spatial dimensions, study their theoretical properties, develop adaptive meshing strategies based on the medium’s inhomogeneity and anisotropy, and establish preconditioning techniques for the resulting linear system to improve the performance of iterative solvers. The expected outcome of the project is improved simulation capabilities for electromagnetic wave propagation, in terms of • performance of numerical methods, including high-order accuracy, scalability, and efficiency, • robustness in regimes including high contrast, high anisotropy, and high frequency, • complexity of mathematical models. Beyond time-harmonic wave-propagation, the development and implementation of HPC quasiTrefftz methods also represents a significant step towards broadening the domain of application of Trefftz methods in general, as (1) they do not require the knowledge of explicit solutions to the governing equations, and (2) they allow for more flexibility in the choice of basis functions. In addition, the proposed work will lead to new scientific contributions at the interface between the fields of mathematics and plasma physics.