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- Title
Mode matching methods for spectral and scattering problems.
- Authors
Delitsyn, A; Grebenkov, D S
- Abstract
We present several applications of mode matching methods in spectral and scattering problems. First, we consider the eigenvalue problem for the Dirichlet Laplacian in a finite cylindrical domain that is split into two subdomains by a 'perforated' barrier. Using rather elementary methods, we prove that the first eigenfunction is localized in the larger subdomain, that is, its |$L_2$| norm in the smaller subdomain can be made arbitrarily small by setting the diameter of the 'holes' in the barrier small enough. This result extends the well-known localization of Laplacian eigenfunctions in dumbbell domains. We also discuss an extension to noncylindrical domains with radial symmetry. Second, we study a scattering problem in an infinite cylindrical domain with two identical perforated barriers. If the holes are small, there exists a low frequency at which an incident wave is almost fully transmitted through both barriers. This result is counterintuitive as a single barrier with the same holes would fully reflect incident waves with low frequencies.
- Subjects
NUMERICAL analysis; PARTIAL differential equations; GALERKIN methods; FINITE element method; BOUNDARY value problems
- Publication
Quarterly Journal of Mechanics & Applied Mathematics, 2018, Vol 71, Issue 4, p537
- ISSN
0033-5614
- Publication type
Article
- DOI
10.1093/qjmam/hby018