T-coercivity and Target Signatures in Computational Wave Propagation

• Applicant: Dr. Martin Halla
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541433971

Wave propagation is ubiquitous in every day life, and many of the related phenomena are modeled by partial differential equations in unbounded domains. A fundamental computational tool to simulate such problems is the complex scaling method, also known as perfectly matched layer (PML) method. This method has been used under a variety of names (analytic dilation, spectral deformation) since the 1970s in molecular physics to compute resonances. In the 1990s Bérenger introduced his PML method for time-dependent electromagnetic wave equations, and soon it was recognized that PMLs are a complex scaling technique. The work of Bérenger has had a huge impact and PMLs have become tremendously popular for all kinds of wave propagation problems, because the additional coding effort is extremely low. However, despite the usage of PMLs for decades, the convergence analysis of these methods is still at its beginning and there exist only a handful of articles on resonances problems and time-dependent equations. Furthermore, a successful adjustment of the PML method for general anisotropic elastodynamics has been an open problem for at least 20 years. Another important aspect in computational wave propagation are eigenvalue problems (EVPs). In particular, many studies on topics such as transmission eigenvalues and classical problems involving conductive media, dispersive media, impedance boundary conditions or resistive sheets lead to nonlinear, nonselfadjoint EVPs. Due to their nonlinear and nonselfadjoint character the analysis and reliable simulation of such problems have not yet been sufficiently well understood. To tackle these challenges we plan to apply a novel set of theoretical tools: T-coercivity techniques. These tools provide a powerful approach to work out the decisive technical details to either prove the convergence of existing methods or to motivate the construction of novel schemes that overcome the obstacles identified in previous attempts. An additional topic connecting the different parts of this project are target signatures for qualitative inverse scattering methods. The inverse methods to be developed rely on the simulation of nonstandard classes of EVPs and furthermore on the simulation of auxiliary scattering problems by means of PMLs.

• DFG Programme: Independent Junior Research Groups
• International Connection: Austria, France, USA
• Cooperation Partners: Dr. Maryna Kachanovska; Hugo Lourenco-Martins, Ph.D.; Professor Peter Monk, Ph.D.; Dr. Markus Wess