High order approximation of transmission eigenvalues

This project concerns the study of high order numerical methods to efficiently approximate transmission eigenvalues, or frequencies, with high accuracy. The transmission eigenvalue problem is of main interest in the analysis of well-posedness of the inverse problem to the Helmholtz scattering problem. The Helmholtz problem describes the propagation of waves. In the inverse problem, one is interested to determine the properties of the scatterer only from the measured scattered wave. The transmission eigenvalues can also be used to obtain bounds on the refractive index of the scatterer, which describes how much the wave is bend, or refracted, when entering the scatterer. Thereby, the refractive index depends on the density of the scatterer. Hence, the efficient numerical computation of transmission eigenvalues with high accuracy is of great interest for the mathematical theory and for practical applications. Since the transmission eigenvalue problem is a non-symmetric fourth order eigenvalue problem, it is very challenging from the numerical point of view. I will study special nonconforming finite element methods, which are easier to generalize to high polynomial degrees than conforming finite elements. So far only lower order methods have been considered, so that these higher order methods will lead to a major advancement in the computation of transmission eigenvalues. Since high order methods require additional mesh refinement close to corners of the domain, I will study adaptive algorithms that automatically choose the local mesh size and local polynomial degree based on some error estimator. For that purpose I plan to derive two different error estimators, one will be based on calculating jumps of normal derivatives across element interfaces, and another one will rely on a recently presented technique based on solving small local auxiliary problems in the mixed finite element space of Hellan-Herrmann-Johnson. Both error estimators will lead to an automated algorithm to compute transmission eigenvalues efficiently with high accuracy. Since most of the time to calculate transmission eigenvalues is spent in solving the discretized matrix eigenvalue problem, I will develop an efficient stopping criterion for iterative eigenvalue solvers. This will additionally speed up the numerical computation of transmission eigenvalues.

In this Lise-Meitner-Project we studied adaptive numerical methods of high order, to calculate frequencies with high precision. Due to the offer of a professor position at the University of Bonn, the project ended early.