An FMM for waveguide problems of 2-D Helmholtz’ equation and its application to eigenvalue problems

Stability of boundary element methods for the two dimensional wave equation in time domain revisited

“…We note that a similar observation has been made in Chappell et al [15,16] qualitatively. It is well-known that the eigenvalues of the frequency domain BIE can be classified into true and fictitious eigenvalues [25,26]. In the exterior problems, the true eigenvalues are with negative imaginary parts, while the behaviour of the fictitious eigenvalues depend on the particular choice of integral equations.…”

“…There exist various possibilities of integral equation for transmission problems on Γ, of which we consider the following four [25,26]: PMCHWT…”

“…In this paper we propose to resolve this difficulty by carrying out the required stability analysis in frequency domain. Namely, we convert the stability analysis for BIEMs in two dimensional wave equation to a non-linear eigenvalue problem similar to those for the Helmholtz equation and solve it with SSM using techniques proposed in Misawa et al [25,26]. This approach has an additional benefit of making the relation between eigenvalues of the approximated integral operators in frequency domain and the stability clearer, thus providing new intuition into the subject.…”

Boundary Integral Equations for Calculating Complex Eigenvalues of Transmission Problems

“…Many efforts have been devoted to numerical and approximate solvers for resonance problems, e.g., [2,3,4,5,6] to mention just a few. The present authors have been interested in solving resonance problems with Boundary Integral Equation Method (BIEM) formulated with Green's functions [7]. This method solves resonance problems by finding frequencies at which the discretized homogeneous BIEs have non-trivial solutions.…”

“…The SSM is a non-iterative algorithm which determines eigenvalues within a given contour γ in the complex plane using contour integrals defined on γ, as does another well-known eigensolver FEAST [14]. As a matter of fact, the authors have developed an FMM for Neumann waveguide problems for the Helmholtz equation in 2D, and solved resonance problems successfully in [7] with the help of the SSM. The third difficulty is caused by the difference between the eigenvalues of the BVP and those of the BIE which may include eigenvalues irrelevant to the original BVP.…”