# Non-homogeneous Maxwell's Equations

I’ve been trying to implement solvers for Maxwell’s equations (both scattering and eigenvalue problems) in 2 and 3 dimensions. Everything worked fine in 2D (including complex domains, eigenvector sources and PML).
However, in 3D, even small problems seem to take much longer to solve than I would expect. I’m guessing I’m doing something wrong.

I attached a simple example:
https://ngsolve.org/media/kunena/attachments/1386/test3d.py (no PML and only 2 materials) to show what I’m doing. Should I be using a different solver or preconditioner?

Any help is appreciated!

Attachment: test3d.py

Hello,

one big difference is that in 3D you have “many” negative eigenvalues, evaluating the biform with any gradient field leads to the negative sign.

A simple method is the “shifted Laplacian” preconditioning: Setup a second, artificial biform with volume damping, build the preconditioner for the artificial problem, and use it to solve the original equation:

``````apre = ngsolve.BilinearForm(fes)
apre += p * (ngsolve.curl(u) * ngsolve.curl(w)) * ngsolve.dx
apre += -(1+1j)*(k0 ** 2) * q * (u * w) * ngsolve.dx

pre = ngsolve.Preconditioner(apre, "bddc")``````

Be careful to choose the sign of the imaginary part of the L2-coefficient according to damping, when you have also physical damping.

It still takes many iterations, and you may have to play with the damping coefficient.

Some references are: Erlangga, Vuik and Oosterlee

Best, Joachim

Attachment: test3d.py

Thanks for the input, Joachim! I’ll go thru the references you indicated and see whether I can improve the code. Our use case is for topological optimization, so any gains in a single run is a big advantage. We currently use COMSOL, but we’d love to use an open-source alternative!