Convergence (FEM)
How an Adaptive FEM Solver Decides It Has Converged
A finite element solver does not know in advance how fine the mesh must be. It begins with a coarse initial mesh that respects the geometry and the requested frequency, solves Maxwell equations on that mesh, and then evaluates a posteriori error estimators that flag where the computed fields are least self-consistent, typically through the discontinuity of the tangential field across element faces and the local residual of the curl-curl equation. Those high-error regions, typically sharp metal edges, dielectric interfaces, and tight gaps, receive a fresh batch of smaller tetrahedra in the next adaptive pass. The solve-estimate-refine loop repeats, and after each pass the solver compares the new port S-parameters against the previous pass.
The comparison metric, delta-S, is the largest magnitude difference across the full S-matrix between consecutive passes. When delta-S falls below the user tolerance, and stays below it for the required number of consecutive passes, the run is declared converged and the mesh is frozen for the frequency sweep. Choosing the adaptive frequency well matters as much as the tolerance: meshing at the highest frequency of interest, where wavelengths are shortest and element density requirements are most demanding, keeps the elements-per-wavelength density adequate across the rest of the sweep. The exception is a sharply resonant or high-Q structure, where the fields are most mesh-sensitive at the resonant frequency, so meshing at or near the band edge containing the resonance is the safer choice.
Two distinct strategies drive the refinement. The h-refinement approach shrinks element size h while holding the basis polynomial order fixed, and the discretization error decreases as h to the power (p+1). The p-refinement approach holds the elements fixed but raises the polynomial order p, which for smooth fields produces exponential error reduction. Modern solvers blend both, and they treat adaptive mesh generation as the engine that makes convergence practical without forcing the engineer to hand-build a fine mesh everywhere.
Convergence Equations
ε ∝ hp+1
h = element edge length, p = basis polynomial order
Error (p-refinement, smooth fields):
ε ∝ e−c·p
exponential decay; c is a problem-dependent constant
Convergence criterion (delta-S):
ΔS = maxi,j | Sij(n) − Sij(n−1) | < τ
n = adaptive pass index, τ = tolerance (e.g. 0.02)
Mesh growth target: refine 10% to 30% of tetrahedra per pass to balance stability against pass count.
Convergence Behavior by Adaptive Pass
| Pass | Tetrahedra | Typical ΔS | Solve time | Interpretation |
|---|---|---|---|---|
| 1 (seed) | 1k to 10k | n/a | Seconds | Coarse baseline, do not trust |
| 2 | 3k to 30k | 0.05 to 0.2 | Tens of seconds | Improving, still mesh-limited |
| 3 | 5k to 50k | 0.02 to 0.05 | Minutes | Approaching tolerance |
| 4 | 8k to 80k | 0.01 to 0.02 | Several minutes | Converged at τ = 0.02 |
| 5+ | 10k to 100k+ | < 0.01 | Tens of minutes | Diminishing returns, check cost |
Frequently Asked Questions
What delta-S value should I set for a converged HFSS-style solution?
A maximum ΔS of 0.02 between consecutive adaptive passes is the common default and is adequate for most passive structures such as filters, couplers, and transitions. For high-Q resonators or precise resonant-frequency extraction, tighten to 0.01 or 0.005, because a small S-magnitude change near a deep notch can still mean a meaningful frequency shift. Require the criterion to hold for at least two consecutive passes, and mesh at the highest frequency of interest so the mesh is also adequate lower in the sweep.
Why does my FEM solution oscillate instead of converging monotonically?
Adaptive refinement is not guaranteed monotonic: each pass adds tetrahedra where the error estimator flags the steepest gradients, which can locally over-correct and swing the result before it settles. Oscillation is common at sharp metal edges, corner field singularities, and in high-Q cavities. Limit refinement to 10% to 30% of tetrahedra per pass, use higher-order basis functions on curved surfaces, seed manual refinement at known singularities, and mesh at a band-edge frequency. Persistent oscillation often points to sliver faces or overlapping geometry producing degenerate elements.
How does FEM convergence differ from the convergence used in digital predistortion?
They share a word but solve different problems. FEM convergence is spatial: it asks whether the mesh is fine enough that the numerical Maxwell solution stops changing, measured by ΔS between passes and governed by element size h and order p (error scales as hp+1). Convergence in digital predistortion is an adaptive-filter question about whether the linearization coefficients have settled so ACLR or EVM stops improving. One is a one-time solver setting before you trust a simulation; the other runs continuously in hardware to track a changing amplifier.