There is a need for simulating unbounded physical domains in many fields, e.g. problems in elecromagnetism or more generally in the study of waves. Even after application of reduction techniques, these physical domains are typically very large leading to a high computational cost for many direct discretization methods. In this project, we take an approach where the domain is treated as an infinite domain, leading to a nonlinear eigenvalue problems [1]. The nonlinear eigenvalue problem is highly structured and we design iterative methods for this structure. New approaches are developed from a numerical linear algebra perspective, e.g. using Arnoldi’s method and modern iterative methods for linear systems, as well as a perspective of numerical methods for partial differential equations. This includes the study and development of preconditioned methods, and preconditioners whose convergence are characterized in theory and practice.
[1] E. Jarlebring, G. Mele and O. Runborg. The waveguide eigenvalue problem and the tensor infinite Arnoldi method. SIAM J. Sci. Comput., 2017Iterative eigenvalue algorithms in unbounded domains
