What is the convergence criterion for a finite element electromagnetic simulation?

The convergence criterion for a finite element method (FEM) electromagnetic simulation determines when the solution is sufficiently accurate and further mesh refinement will not significantly change the results. In Ansys HFSS (the most widely used FEM EM solver), the primary convergence criterion is the maximum delta-S: the maximum change in any S-parameter magnitude between successive adaptive mesh refinement passes. Default: delta-S < 0.02 (2% change). For high-accuracy work: delta-S < 0.01 or delta-S < 0.005. The adaptive mesh process: (1) HFSS generates an initial mesh based on the geometry (surface and volume meshes using tetrahedral elements, seeded at approximately lambda/5 element size). (2) Solves Maxwell equations on this mesh at the solution frequency. (3) Calculates the electric field error estimator for each element. (4) Refines the mesh by subdividing elements with the highest error. (5) Re-solves on the refined mesh. (6) Compares S-parameters between the current and previous solution. If max(|delta_S_ij|) < convergence target (e.g., 0.02): the solution has converged. If not: repeat steps 3-6. Typically, 3-8 adaptive passes achieve convergence for well-defined problems. Additional convergence checks: (1) Energy-based convergence: the stored electromagnetic energy should change by <1% between passes. (2) Field convergence: maximum field magnitude change <5% in critical regions. (3) Manual verification: compare results with known analytical solutions (e.g., a rectangular waveguide mode should match theoretical cutoff frequency within 0.1%).