# Finite element method for high-frequency electromagnetics

I am writing a project about the Finite element method for use in high-frequency solutions of Maxwell's equations. This could be for use in antenna design and similar. I have some trouble understanding how to choose the form of the maxwells equation to solve. It seems that for high-frequency problems people use the second order equations, the one for only one field. $$\nabla \times \mu^{-1} \nabla \times E + \epsilon \frac{\partial^2 E}{\partial t^2} = 0$$ Others solve the first-order equations for both the electric and magnetic fields. This seems to often be for use in eddy current problems, like electric motors and transformers.

$$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$ $$\nabla \times \mathbf{B} = \mu \left( \mathbf{J} + \epsilon \frac{\partial \mathbf{E}}{\partial t}\right)$$

Is this because it is simpler to solve the second-order equation, but you need more information for low-frequency problems like electric motors?

I have mostly used the books: "The finite element method in electromagnetics" by Jian-ming Jin, and "Computational Electromagnetics" by Thomas Rylander, Par Ingelstrom, and Anders Bondeson.

Does anyone have any insight into this?

Using the typical expansion functions (1-forms/edge-elements for E, and 2-forms/facet-elements for B) the formulations are basically the same after spatial discretization and you'd expect more or less the same accuracy. I do think they express slightly different opinions about time integration.

The mixed E/B formulation nudges you in the direction of leapfrog integration, leading towards something FDTD-like but with unstructured gridding instead of sugarcubing. This formulation is not very popular because it is both conditionally stable (like FDTD's CFL condition, but worse because you have less control over element shape) and yet it still requires an implicit solve by a non-diagonal mass matrix (although it is well conditioned / spectrally equivalent to identity, it's still an annoyance).

The all-E formulation nudges you in the direction of Newmark integration, which is unconditionally stable (not limited by element size/shape). This method also requires an implicit solve and the system is more poorly conditioned (it's a weighted sum of the the stiffness and mass matrices, so it inherits unboundedness from the curl-curl operator).

Of course, there's nothing that says you have you use these particular time integrators. You can always submit these systems to a black-box ODE solver (Runge-Kutta, etc).

Since you mention antenna modeling, another method that should be considered is frequency domain FEM, based on the vector wave equation $$\nabla \times \mu_r^{-1} \nabla \times \vec E - k^2 {\epsilon}_r \vec E = 0$$. These systems are rather difficult to solve/precondition: oscillatory behavior like Helmholtz, compounded with the complexity of faithfully preconditioning the nullspace of the curl operator. But, they can be readily hybridized with other frequency domain techniques (modal expansions, method of moments, etc). These are much more accurate termination conditions than ABC's/PML's/etc, which can be important because many quantities of engineering interest (gain, RCS) are derived from the far-field.

As an aside, I'd recommend Jian-Ming Jin's "Finite Element Analysis of Antennas and Arrays", it's excellent (though a little more advanced). It has in-depth coverage of the time-domain Newmark method and frequency-domain methods/hybrids.

• Is there something similar for the frequency domain. In the book "Computational Electromagnetics” he solves a system using the frequency domain version of the first order equations. The book supplies some MATLAB code that solve a structure using these. I have modified his code to solve, the same system using the vector wave equation. It seems to solve the problem with the same accuracy. Most papers mostly seem to use the vector wave equation for the E field. Maybe having both the electric and magnetic field would allow you to have more complex materials and boundaries?
– bbch
Apr 6, 2021 at 19:26
• My intuition in the frequency domain is to favor the 2nd order E system (vector wave equation) over the 1st order E/B system (Maxwell equations). Though they should have the same accuracy, the VWE should have about half the number of unknowns. You can look at the VWE as essentially the reduced/Schur complement system that arises from eliminating B all the way back at the formulation level. Apr 6, 2021 at 21:27
• Regarding material handling, there could be slight advantage for the E/B system, mainly because the VWE needs the inverse of mu which can be awkward. This is not a major concern, mainly because exotic magnetic media are less common than exotic dielectric media. Ferrite would be the most notable exception, but it's easy enough to invert that relationship (the Polder tensor). Apr 6, 2021 at 21:29