NLEVP: A Collection of Nonlinear Eigenvalue Problems
ACM Transactions on Mathematical Software (TOMS)
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Guided and leaky modes of planar dielectric waveguides are eigen-solutions of a singular Sturm-Liouville problem. These modes are also the roots of a characteristic function which can be found using several methods that have been introduced in the past. However, the evaluation of the characteristic function suffers from numerical instability, and hence it is often difficult to find all modes in a given range. Here, a new variational technique is introduced. After discretization the variational formulation leads to a nonlinear eigenvalue problem in the propagation constant. Typically, this type of problem must be solved with a Newton-like procedure. Thus the modes that can be found depend on a judicious choice of the initial guess, which is normally not available. We show that after a change of variables the nonlinear problem can be transformed to either a quadratic or a quartic eigenvalue problem, depending on the waveguide structure. Using the companion matrix, such a problem can in turn be converted into a linear eigenvalue problem. The advantage is that now one can use standard numerical algorithms, such as the QR iteration or Arnoldi methods, which guarantee that all important modes are found. Because the resulting matrices are sparse, we use an Arnoldi method to compute the eigenvalues. Using the physical properties of the modes, we demonstrate how to compute the shifts that enable us to accelerate the convergence of the iterations to the desired portions of the spectrum. In addition, by using high-order finite elements, the resulting solutions can be made extremely accurate. We show by example that the classical error estimates for standard and generalized eigenvalue problems hold for the polynomial eigenvalue problems derived here. In addition, it is observed that the FE computed eigenvalue become polluted for large frequencies. We show that this pollution is consistent with the known error estimates. Numerical examples are presented.