A new computational method for electromagnetic problems required far fewer iterations than a Robin transmission-condition approach in a series of numerical tests. The largest reported reduction was 9.53-fold for a METIS-partitioned waveguide, while the calculated fields matched the comparison solutions in the cases tested.
The work asks whether a finite-element domain-decomposition solver can provide inexpensive, rapid and stagnation-free iterative solutions across varied settings while retaining massive parallelism and scalability. The proposed solver focuses on the global interfaces between subdomains, where tangential fields and normal fluxes must be matched.
A second channel for matching fields
The proposed FETI-DP-4λ method uses a mixed formulation that represents both the electric field and magnetic flux. Its two-channel transmission condition pairs tangential fields with normal fluxes through separate Faraday and Ampère-Maxwell channels, using four Lagrange multipliers at each interface to connect the subdomains.
The comparisons used identical conformal tetrahedral meshes and solver settings. Local subdomain systems were factorized with a sparse direct solver, while the global interface equation was handled by unpreconditioned GMRES, an iterative algorithm that repeatedly reduces a residual, or measure of the remaining numerical mismatch. The reported tests used a restart parameter of 1,200 and a relative residual tolerance of 10−10.
The mixed design carries a computational cost: it stores both field types and makes the interface traces and global system larger than electric-field-only versions. The paper states that it adds no subdomain solves, however, and that the extra work per iteration rises by a bounded factor while iteration counts fall by larger factors in the reported cases.
The gains appeared across several partitions
In the first test, a WR-90 waveguide operating at 8.2 GHz was divided into 40 uniform slabs, each 6 millimetres wide, creating 39 cross-section interfaces. At a 10−6 residual tolerance, the proposed transmission condition reached the target in 55 GMRES iterations, compared with 188 for Robin. At 10−10, it took 68 iterations against 301, figures reported as reductions by factors of 3.42 and 4.42.
The lower iteration counts did not come with a reported loss of accuracy in that uniform case. Both transmission conditions agreed with the analytical solution to approximately 99.25%, and the electric-field relative L2 difference between the proposed and Robin solutions was 3.2 × 10−7. At the tighter tolerance, the reported wall-clock time was 380.7 seconds for Robin and 189.3 seconds for the proposed method, an approximately twofold speedup.
The method was also tested on a less evenly divided WR-90 guide, split into 30 slabs ranging from 0.8 to 18.8 millimetres wide. There, the proposed condition converged in almost nine times fewer iterations than Robin, with a relative L2 difference of 0.0001%. Additional cases covered ridge-loaded and L-shaped waveguides, as well as a METIS-partitioned WR-90 guide. At 10−10, the reported reductions were 3.39-fold for the ridge-loaded case and 9.53-fold for the METIS case, with no loss of accuracy; similar reductions were reported for L-shaped cases using 40 and 100 subdomains.
A filter model showed a similar pattern
The tests extended to an X-band bandpass waveguide filter simulated at 8.4 GHz and divided into 110 subdomains. To reach a 10−10 residual, the proposed transmission condition required 180 GMRES iterations, compared with 1,020 for Robin. Across the METIS partitions tested, from 50 to 310 subdomains, the improvement was reported as nearly sixfold.
The authors also examined how closely fields matched across subdomain boundaries. In the METIS-partitioned waveguide at a 10−6 stopping tolerance, the worst normal-flux jump was 9.1 × 10−7 with Robin and 2.2 × 10−12 with the proposed condition. The worst tangential-electric-field jump was 1.4 × 10−8 with Robin and 1.2 × 10−10 with the proposed condition. At the looser 10−3 tolerance, the proposed condition was reported to give more accurate solutions.
What the numerical tests establish
The reported evidence comes from deterministic numerical comparisons using mixed E-B finite elements, conformal tetrahedral meshes, sparse direct subdomain factorizations and unpreconditioned GMRES. The cases include uniform and irregular WR-90 partitions, ridge-loaded and L-shaped waveguides, a METIS-partitioned WR-90 guide and an X-band filter, under the solver settings and residual tolerances described above.
The authors interpret these results as evidence that the proposed transmission condition can reduce global-interface iterations by up to several-fold, preserve solution accuracy, lower interface field and flux jumps, and support robust, accurate, massively parallel and scalable iterative domain-decomposition solvers. Those conclusions are tied to the waveguide and filter scenarios, meshes and solver settings used for validation.
The document is identified as arXiv:2608.25041v1 [physics.comp-ph] and dated 25 August 2026. The manuscript reports partial support from the Ohio State President’s Research Excellence Program and the Ohio Supercomputer Center Grant PAS-0061.
Paper data and sources
Original title: Rapidly Convergent Finite-Element Domain Decomposition Method With Two-Channel Transmission Conditions
Authors: Furkan Şık, Fernando L. Teixeira, Balasubramaniam Shanker
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
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