Research Projects
Unbounded Domains and Absorbing Boundaries
Many wave propagation problems are posed on domains that are effectively unbounded. Since finite element methods require finite computational domains, artificial boundaries have to be introduced without creating spurious reflections.
My research focuses on Perfectly Matched Layers (PMLs), infinite elements, and related absorbing boundary techniques for time-domain and frequency-domain wave propagation problems, including anisotropic and dispersive media.
This work combines mathematical analysis with efficient finite element discretizations for large-scale simulations.
Fast Time-domain Solvers
Time-domain simulations of acoustic and electromagnetic waves often lead to extremely large systems of ordinary differential equations. Efficient explicit time-stepping methods require specially designed spatial discretizations with favorable mass matrices.
My work focuses on high-order finite element and dual-cell methods that enable efficient explicit solvers while maintaining high accuracy. Applications include large-scale electromagnetic simulations.
Software implementations are available in NGSolve.
Resonance Problems
Resonance phenomena are governed by eigenvalue problems arising from wave equations on open domains. These problems are computationally challenging due to their size and non-Hermitian structure.
My research addresses efficient Krylov methods, nonlinear eigenvalue algorithms, and time-domain techniques for computing resonances in acoustic and electromagnetic systems.
Several algorithms are accompanied by open educational material in the form of interactive Jupyter Books.
Scientific Software Development in NGSolve
Most numerical methods developed in my research are implemented in the open-source finite element library NGSolve.
My contributions include implementations of perfectly matched layers, infinite elements, dual-cell methods, and research prototypes for large-scale wave propagation and eigenvalue problems.
Alongside the software development, I maintain interactive Jupyter Books that document algorithms and demonstrate their practical use.