shift-and-invert eigensolver

[stevengj]

We have a frequency-domain solver that computes the fields from a given current source at a given frequency. It would be nice to be able to iterate this — using the resulting fields as the source in a subsequent frequency-domain solve — in order to implement the "shift-and-invert" power iteration to compute resonant modes.

(In the longer run, we could also do Arnoldi, Rayleigh-quotient iteration, etcetera… If we go that route we should probably just call ARPACK or similar.)

[stevengj]

As explained in the attached notes, I think we actually want to use the D and B fields (not E and H) as current sources in a shift-and-invert power iteration.

freq-domain.pdf

(See also these notes for more background on this formulation of Maxwell's equations.)

Updated: Reviewing the Meep source code, I realized that the linear system we are solving in Meep is already expressed in terms of D and B, so this formulation is especially easy to adapt to a power iteration.