Every electromagnetic simulator solves Maxwell's equations. They differ in how. Three families dominate production work, and each makes a different bargain between accuracy, the geometry it handles gracefully, and the memory and time it burns. If you are new to the topic, start with what is electromagnetic simulation?.
FDTD
FDTD lays a grid over space and marches time forward, with each cell updating from its neighbors. A single pulse reveals the response across a whole band of frequencies at once. It is simple and massively parallel, but the rectangular grid staircases curved shapes, accuracy is fixed at second order, and open problems need an artificial absorbing boundary. Numerical dispersion also builds up across electrically large runs, so accuracy erodes as the structure grows.
FEM
FEM fills the whole domain with a mesh of small elements that conform to the geometry, then solves one large sparse system at a single frequency. It is the most flexible of the three on complex shapes and inhomogeneous materials, but it has to mesh the empty space around the object and cap it with an absorbing layer (a PML), and you repeat the solve for every frequency in the band. At high frequency it also suffers the pollution effect: the mesh must grow faster than the wavelength shrinks just to hold accuracy steady.
MoM
The method of moments places unknowns only on surfaces, and lets a Green's function handle radiation exactly, so no empty space is meshed and no artificial boundary is needed. That makes it ideal for antennas and open scattering. Every unknown couples to every other, so the matrix is dense, which is costly to solve directly. Fast multipole methods bring that down to effectively linear cost, which is what makes large problems practical. It largely escapes the pollution effect that hits the volume methods, but standard surface formulations bring conditioning problems of their own: spurious interior resonances at certain frequencies and a low-frequency breakdown, both of which need specialized formulations or preconditioning.
Side by side
| FDTD | FEM | MoM | |
|---|---|---|---|
| Works in | time domain | frequency domain | frequency domain |
| Broadband result | one run | one solve per frequency | one solve per frequency |
| Complex geometry | staircased on a grid | conforming mesh | conforming mesh (surfaces) |
| Inhomogeneous materials | native | native | volume or hybrid only |
| Open-domain radiation | needs absorbing layer | needs absorbing layer | handled analytically |
| Meshes empty space? | yes | yes | no |
| Linear system | none (explicit) | large, sparse | dense |
How to choose
- Need the broadband response of a moderate structure? FDTD gives the whole spectrum in one run.
- Working with intricate geometry and inhomogeneous materials, or chasing resonances? FEM is the most forgiving.
- Designing antennas or open scattering, especially metallic structures? MoM avoids meshing the air entirely.
All three trade accuracy against cost on hard geometry. The difficulties that strain them are the same: sharp corners, high contrast, inhomogeneous media, and open radiation. That is exactly where the order of accuracy starts to matter, the subject of why convergence order matters.
Planck Labs is building an integrated design platform for high-frequency 3D devices, with a high-order electromagnetic solver at its core.