### Problem-adapted discretisations of wave equations (Ilaria Perugia)

[P7, Perugia]

focuses on the analysis and implementation of numerical methods for wave propagation problems both in frequency and in time domain, and for the stationary Schrödinger equation.
The finite element approximation of wave propagation models which feature oscillatory solutions (time-harmonic acoustic, electromagnetic and elastic wave problems, quantum mechanic systems) is challenging, due to the strong requirements on the mesh resolution in order to obtain accurate approximations. Our aim is to reduce the complexity in such problems by further developing methods which employ problem-adapted, oscillatory basis functions which possess better approximation properties, as compared to standard polynomial ones. For multimaterial scattering problems, the coupling with boundary elements will be studied in order to accurately impose the far-field condition. For wave problems in time-domain, we will combine the use of problem-adapted basis functions with problem-adapted meshes of the space-time domain constructed following the so-called tent-pitching approach, a front-advancing technique in time based on the local wave velocity. Block-based numerical simulations will be integrated within the

SFB software platform

.
[P7, Perugia]

collects the SFB activities on numerics of wave problems. The coupling of finite elements and boundary elements for time-harmonic scattering will be coordinated with
[P6, Melenk],
the study of finite element methods for the stationary Schrödinger equation with
[P1, Arnold],
and that of space-time discretisations of wave problems in time-domain with
[P10, Schöberl].
### Members of this project part

Ilaria Perugia (Project Leader)

Francesca Bonizzoni (Postdoc)

Lorenzo Mascotto (Ph.D. Student)

Paul Stocker (Ph.D. Student)