A Preconditioned Adaptive Finite-Element Method (PC-AFEM) for Microwave Modeling

The adaptive finite-element method (AFEM) in microwave modeling follows the heuristic cycle of “solve, estimate, mark, and refine.” In engineering practice, it is crucial to maximize efficiency while maintaining the accuracy of each step in this process. To address this, we propose a preconditioned AFEM (PC-AFEM) approach. This method incorporates a preconditioning technique for Nédélec elements during the “solve” phase, reducing the computational time required for each adaptive pass on the coarsest initial mesh. Additionally, we introduce a novel energy-norm-based error indicator for a posteriori error estimation in the “estimate” phase, which enhances both the reliability and efficiency of element marking. Numerical experiments demonstrate the advantages of the proposed PC-AFEM method: 1) in benchmark studies, the adaptive iteration time is reduced by 44.0% compared with HFSS, with memory usage decreased by 76.5%, while achieving results closer to the reference solution at a comparable computational scale; 2) in classic filter design scenarios, the computational burden of the PC-AFEM is 34.9% lower than that of the state-of-the-art error estimation methods; and 3) in the design of a double-sided substrate-integrated suspended line (DSISL) coupler, the algorithm efficiently generates a high-quality adaptive mesh and calculates its $S$ -parameters, confirming its practical applicability in engineering.