Solving Maxwell eigenvalue problems for accelerating cavities

Abstract

We investigate algorithms for computing steady state electromagnetic waves in cavities. The Maxwell equations for the strength of the electric field are solved by a mixed method with quadratic finite edge (Nédélec) elements for the field values and corresponding node-based finite elements for the Lagrange multiplier. This approach avoids so-called spurious modes which are introduced if the divergence-free condition for the electric field is not treated properly. To compute a few of the smallest positive eigenvalues and corresponding eigenmodes of the resulting large sparse matrix eigenvalue problems, two algorithms have been used: the implicitly restarted Lanczos algorithm and the Jacobi-Davidson algorithm, both with shift-and-invert spectral transformation. Two-level hierarchical basis preconditioners have been employed for the iterative solution of the resulting systems of equations.

Algebraic formulations for the solution of the nullspace‐free eigenvalue problem using the inexact Shift‐and‐Invert Lanczos method

“…References [1,5,6] and references therein. One such application is the analysis of the electromagnetic cavity resonator.…”

“…We assume that a sparse basis C for the null space of A is available, as is the case in several applications; see for instance References [1][2][3] and their references. A relevant feature of the considered setting is that the dimension of the null space is very high, compared with the problem dimension, reaching up to, say, one-third of the entire problem dimension.…”

On a parallel multilevel preconditioned Maxwell eigensolver

“…This algorithm is well-suited since it does not require the factorization of the matrices A or M . In [2,3,4] we found JDSYM to be the method of choice for this problem.…”

A Comparison of Factorization-Free Eigensolvers with Application to Cavity Resonators

“…This is very time consuming even if the conjugate gradient method is applied with a good preconditioner. In most of our experiments the JacobiDavidson algorithm was much more effective than the implicitly restarted Lanczos algorithm as implemented in ARPACK [5] for solving the cavity and other eigenvalue problems [2,3]. In this study we investigate three factorization-free algorithms for solving (13)…”

“…In earlier studies [2,3] we found that for large eigenvalue problems the JacobiDavidson algorithm [4] was superior to the Lanczos algorithm or the restarted Lanczos algorithm as implemented in ARPACK [5]. While the Jacobi-Davidson algorithm retains the high rate of convergence it only poses small accuracy requirements on the solution of the so-called correction equation, at least in the initial steps of iterations.…”