Abstract
In this work, we develop an efficient spectral method to solve the Helmholtz transmission eigenvalue problem in polar geometries. An essential difficulty is that the polar coordinate transformation introduces the polar singularities. In order to overcome this difficulty, we introduce some pole conditions and the corresponding weighted Sobolev space. The polar coordinate transformation and variable separation techniques are presented to transform the original problem into a series of equivalent one-dimensional eigenvalue problem, and error estimate for the approximate eigenvalues and corresponding eigenfunctions are obtained. Finally, numerical simulations are performed to confirm the validity of the numerical method.
Highlights
The internal transmission eigenvalue problem of acoustic wave scattering in inhomogeneous media can be expressed as follows: Find k ∈ C, ω, υ ∈ L2(Ω), Ω ⊂ Rd(d = 2, 3), ω − υ ∈ H2(Ω) such that∆ω + k2n(x)ω = 0, in Ω, (1)∆υ + k2υ = 0, in Ω, (2)ω − υ = 0, on ∂Ω, (3) ∂ω ∂υ − = 0, ∂ν ∂ν on ∂Ω, (4)where n(x) > 0 is the refractive index of the material, ν is the unit outer normal vector, and k is the transmission eigenvalue in the sense of nontrivial solution (ω, υ)
Based on the work in [23], we develop a high-order spectral method based on fourth-order scheme for the transmission eigenvalue problem in polar geometries
We present an efficient spectral-Galerkin algorithm for sovling the transmission eigenvalue problem in polar geometries
Summary
Transmission eigenvalue problem, spectral method, polar condition, Error estimate. Let ψmk be an eigenfuction corresponding to λk(τm), there exist a constant c and a function vmN ∈ span{ψmk N , .
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