Fast integral-equation based solutions of the Laplace eigenvalue problems with mixed boundary conditions

Start: 12/01/2014 - 12:00pm
End  : 12/01/2014 - 1:00pm

Applied Math Seminar

Eldar Akhmetgaliyev (Caltech)


In this talk we present a range of numerical methods which, based on use of Green functions and integral equations, can be applied to produce solution of Laplace eignvalue problems with arbitrary boundary conditions (including, e.g., Dirichlet/Neumann mixed boundary conditions) and in arbitrary domains (including e.g. domains with corners). As part of our presentation we present newly obtained characterizations of the singularities of solutions and eigenfunctions which arise at transition points where Dirichlet and Neumann boundary conditions meet; the numerical methods mentioned above rely on use of these characterizations in conjunction with the novel Fourier Continuation technique to produce solutions with a high order of accuracy. In particular, the resulting method exhibits spectral convergence for smooth domains (in spite of the solution singularities at Dirichlet/Neumann junctions) and prescribed high-order convergence for non-smooth domains.

A point of interest concerns the search algorithm in our eigensolver, which proceeds by searching for frequencies for which the integral equations of the problem admit non-trivial kernels. As it happens, the “minimum-singular- value” objective function gives rise to a challenging nonlinear optimization problem. To tackle this difficulty we put forth an improved objective functional which can be optimized by means of standard root-finding methods.

The methods above were also applied to modal analysis problems in electromagnetics: our calculation of TE and TM modes (eigenfunctions) of the cross sections of specifically designed quadruple-ridged flare horn microwave (astrophysical) antenna have been applied to the problem of optimization of the antenna parameters. The resulting eigensolutions are produced with such high accuracy that it becomes possible to track the eigenvalue/eigenfunction evolution with shape changes even as eigenvalues cross—a capability that is necessary for the antenna-design application, and which existing commercial software packages were not able to deliver.

Other applications will also be mentioned, including methods for evaluation of transmission eigenvalues that arise in the field of inverse problems and computation of Laplace eigenfunctions as a basis for spectral decomposition of functions in the space of square integrable functions—with application to, e.g., highly accurate separation-of-variables solution of time-dependent problems (including diffusion and wave-propagation) in arbitrary, possibly singular spatial domains and with possibly mixed boundary conditions.


KRV 164