Trefftz-based approximation of time-harmonic wave problems

Time-harmonic wave propagation problems arise in many environmental and engineering applications (e.g., in acoustics, optics, hydrodynamics, electrical engineering, medical imaging, geophysics, seismology, structural engineering). Their numerical simulation presents intrinsic difficulties, due to the oscillatory nature of the analytical solutions. This project aims at developing accurate and computationally efficient methods for the simulation of acoustic and electromagnetic time-harmonic wave propagation. The finite element method is the most popular technology for the numerical approximation of problems governed by partial differential equations. In its original version, it is based on piecewise polynomial approximating functions. For time-harmonic wave problems, where the solutions present strong oscillations, the computational cost required in order to obtain accurate approximations is prohibitive. The finite element methods that will be studied in this project, instead of polynomials, make use of approximating functions which depend on the problem (Trefftz functions). Specifically, these functions oscillate with the frequency inherent to the original problem. Being able to locally reproduce the behaviour of the analytical solutions, these methodologies allow to obtain a given accuracy with a lower computational cost, as compared to standard polynomial-based finite element methods. Trefftz methods are at the forefront of the research in numerical approximation of wave propagation problems. Developing numerical techniques with improved performance, as this project aims at, may impact on various simulation-based activities in science and technology, where the need of ``virtual`` tests, as complementary, when not substitute, to expensive experimental tests, is of paramount importance.

This project was concerned with the design and the theoretical analysis of numerical methods for the approximation of time-harmonic wave propagation problems. Time-harmonic wave propagation problems are present in many environmental and engineering applications (e.g., in acoustics, optics, hydrodynamics, electrical engineering, medical imaging, geophysics, seismology, structural engineering), and their numerical approximation is notoriously challenging. In fact, due to the oscillatory nature of their analytical solutions, standard approximation methods deliver accurate approximation only at very high computational cost. For this reason, the focus of this project was on finite element methods based on approximating spaces of Trefftz type, namely on spaces made of functions whose restriction to each mesh element belongs to the kernel of the considered differential operator. Being able to locally reproduce the behavior of the analytical solutions, these methodologies allow to obtain a given accuracy with less degrees of freedom, as compared to standard finite element methods based on polynomial functions. However, this advantage was dimmed by the lack of tools for their efficient numerical treatment. We have studied Trefftz methods combined with two different settings: the discontinuous Galerkin setting, where no interelement continuity is required on the approximating functions, and the virtual element setting, which is a recent generalization of the finite element method to polytopic meshes, where the basis functions are not known in closed form (hence "virtual"). In particular, we have provided numerical tools, such as adaptive refinement strategies, to improve the computational efficiency of the more established Trefftz-discontinuous Galerkin methods, and we have introduced a novel Trefftz-virtual element method. The relevance of the latter is two-fold: on the one hand, the new method has shown that the virtual element setting can be extended beyond its original polynomial paradigm, and can accommodate problem-adapted virtual spaces; on the other hand, it has been assessed that, in combination with a suitable variational setting and a suitable numerical treatment, it is possible to significantly improve the efficiency of Trefftz methods, and to extend their applicability. Other outcomes of this project include the design of a method for coupling finite elements and boundary elements for time-harmonic acoustic scattering in heterogeneous media, and the study of model order reduction methods based on rational interpolation for frequency-parametrized wave propagation problems.