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Rapidly Convergent Finite-Element Domain Decomposition Method With Two-Channel Transmission Conditions
Furkan Şık, Fernando L. Teixeira, Balasubramaniam Shanker
TL;DR
Large-scale Maxwell finite-element systems challenge conventional solvers because existing interface conditions can stagnate on evanescent modes and difficult partitions. The paper introduces a two-channel FETI-DP method that separately enforces tangential-field and normal-flux continuity, and experiments report drastically reduced iteration counts with preserved accuracy across diverse settings.
Problem
Electrically large Maxwell problems create systems with millions or billions of unknowns, while Robin and second-order transmission conditions can converge slowly or stagnate on challenging evanescent-wave interfaces.
Method
The proposed FETI-DP-4λ method constructs separate Faraday and Ampere-Maxwell interface channels from a mixed E-B formulation to enforce tangential-field and normal-flux continuity.
Results
Across diverse three-dimensional geometries, partitions, discretizations, and subdomain configurations, the method drastically reduces GMRES iterations relative to Robin and SOTC approaches without loss of accuracy.
Takeaways & Limitations
Direct enforcement of both interface continuities reduces field and flux jumps by orders of magnitude while preserving solution accuracy across the tested electromagnetic problems.
Abstract
from arXiv · showhide
A novel dual-primal finite element tearing and interconnecting (FETI-DP) domain decomposition method (DDM) is introduced for solving Maxwell's equations. The proposed method is built upon a two-channel transmission condition (TC) enforcing both tangential-field and normal-flux continuity across subdomain interfaces. Two interface channels are separately constructed from the Faraday and Ampere-Maxwell equations. Simultaneously enforcing tangential-field and normal-flux continuity reduces interface jumps by several orders of magnitude relative to the widely used Robin TC. The resulting global interface equation exhibits markedly improved iterative convergence. Numerical experiments across different partitioning strategies, subdomain counts and volumes, mesh resolutions, and geometries demonstrate drastic reductions in iteration counts, while accuracy, massive parallelism, and scalability are preserved.
I. INTRODUCTION
Large electromagnetic finite-element problems produce enormous sparse systems, while existing transmission conditions can converge slowly or stagnate on difficult subdomain interfaces. The proposed two-channel condition separately handles tangential fields and normal fluxes, improving convergence and preserving accuracy across extensive experiments.
- Electrically large Maxwell problems can generate sparse systems with millions or billions of unknowns, straining direct and iterative solvers.
- Robin transmission conditions damp propagating interface-error modes but cannot simultaneously damp TE- and TM-type evanescent families with one coefficient.
- Difficult geometries and deep-evanescent waves can slow or stagnate Robin convergence, while second-order conditions require corner penalties and lose damping effectiveness for deep-evanescent waves.
- The proposed two-channel condition separately constructs Faraday and Ampere-Maxwell channels to enforce tangential-field and normal-flux continuity across interfaces.It uses four Lagrange multipliers per interface while retaining dual-primal corner treatment.
- Extensive experiments across geometries, partitionings, discretizations, and subdomain configurations show drastically fewer GMRES iterations than Robin and SOTC approaches without loss of accuracy.
II. FORMULATION
The formulation retains mixed electric and magnetic flux variables and derives a two-channel interface condition from the Faraday and Ampere-Maxwell equations. The resulting FETI-DP-4λ system uses dual-primal corner treatment, subdomain condensation, and a GMRES-solved global interface equation.
- Mixed E-B discretization: The mixed formulation retains E in H(curl) and B in H(div), expanded with Whitney edge and face bases rather than reducing Maxwell’s equations to a second-order wave equation.The subdomain unknown is ordered from discrete magnetic-flux and electric-field coefficients.
- Two-channel transmission condition: The two-channel transmission condition assigns one channel to each curl equation, pairing tangential traces with normal-flux terms on every interface.The Faraday channel uses n × E with n(n · B), while the Ampere-Maxwell channel uses n × H with n(n · D).
- Interface coupling: Each interface carries four Lagrange multipliers because two independent channels are enforced from both neighboring subdomains.The multipliers act as incoming interface data, while the two sides are coupled through neighbor-side traces and orientation operators.
- Interface operators: The formulation assembles tangential and flux pairings into local block systems, with the mixed setting representing magnetic normal traces directly and electric normal flux through a surface grad-div operator.The electric and magnetic flux operators are identical on the two sides and introduce no auxiliary interface unknown.
- FETI-DP-4λ assembly: Dual-primal partitioning separates non-corner coefficients from globally assembled primal corner-edge coefficients before subdomain factorization and Schur reduction.One sparse factorization per subdomain condenses the system onto the primal corner variables and interface multipliers.
- Iterative solution: GMRES solves the global interface equation using subdomain-level solves and one small corner solve, followed by local back-substitution for field and flux coefficients.The added channel increases local-system and multiplier-vector sizes, but requires no additional subdomain solves; the per-iteration cost rises by a bounded factor.
III. NUMERICAL EXPERIMENTS
The numerical study evaluates the proposed method against established transmission conditions on 3-D Maxwell problems using matched meshes and solver settings. Experiments span varied geometries, partitionings, discretizations, and subdomain configurations.
- The proposed and comparison methods use identical conformal tetrahedral meshes and solver settings across a sequence of 3-D examples.Subdomain problems use mixed E-B variables and sparse direct factorization; unpreconditioned GMRES solves the global interface equation to 10^-10 relative residual tolerance.
- The experiments cover waveguide geometries, partitionings, discretizations, subdomain counts, and subdomain sizes.
- GMRES convergence histories are reported for irregularly partitioned waveguides.
- GMRES iteration counts are tabulated for the uniformly partitioned waveguide.
A. WR-90 Waveguide: Uniform and Irregular Partitionings
For WR-90 waveguides, the proposed transmission condition preserves solution accuracy while substantially reducing GMRES iterations under both uniform and irregular partitionings.
- Uniform partitioning: 3.42× and 4.42× fewer GMRES iterations are required than with Robin TC at 10^-6 and 10^-10 tolerances, respectively.For the uniform 40-slab case, both methods agree with the analytical solution to approximately 99.25%.
- Uniform partitioning: ∼2× lower wall-clock time is achieved at 10^-10 tolerance, decreasing from 380.7 s to 189.3 s.
- Irregular partitioning: Almost 9 times fewer iterations are obtained than with Robin TC for 30 irregular slabs with widths ranging from 0.8 to 18.8 mm.The two solutions differ by only 0.0001% in relative L2 norm.
- Irregular partitioning: Thin, closely spaced irregular slabs couple through evanescent modes that Robin TC leaves undamped and that degrade SOTC damping.
B. Ridge-Loaded, L-Shaped, and METIS-Partitioned Waveguides
Additional ridge-loaded, L-shaped, and METIS-partitioned waveguides test the method under geometric irregularity, bends, corner edges, and varying subdomain counts. The proposed condition consistently reduces GMRES iterations without loss of accuracy.
- Geometries and partitionings: The tested geometries include a ridge discontinuity, a right-angle bend, and irregular METIS partitioning.
- Ridge-loaded and METIS-partitioned waveguides: GMRES convergence histories are shown for ridge-loaded and METIS-partitioned waveguides, with the METIS mesh partition displayed in an inset.
- L-shaped waveguide: GMRES convergence histories are shown for the L-shaped waveguide with 40 and 100 subdomains.
- Ridge-loaded and METIS-partitioned waveguides: 3.39× and 9.53× fewer GMRES iterations are achieved than with Robin TC for ridge-loaded and METIS-partitioned waveguides, respectively.The ridge-loaded case uses 40 subdomains; the METIS case uses 16 irregular subdomains with 37 interfaces and multiple corner edges.
- L-shaped waveguide: Similar iteration reductions are observed for the L-shaped waveguide partitioned into both 40 and 100 subdomains.
C. X-Band Bandpass Waveguide Filter
The X-band bandpass filter provides a demanding multiscale test with strong evanescent coupling and many corner edges after METIS partitioning. The proposed condition substantially accelerates convergence across tested partition sizes.
- Filter geometry and convergence: 180 versus 1020 GMRES iterations are required to reach 10^-10 tolerance, a nearly sixfold reduction over Robin TC.The filter operates at 8.4 GHz and is partitioned into 110 subdomains.
- Filter geometry and convergence: The filter combines multiscale features, strong evanescent near-field coupling, and numerous corner edges.These characteristics make it a demanding problem for Robin TC.
- Filter geometry and convergence: Figure 4 presents the METIS-partitioned geometry and the corresponding GMRES convergence histories.
- Filter convergence: Consistent improvement is observed across METIS partitionings with subdomain counts ranging from 50 to 310.
D. Improved Interface Continuity
The two-channel FETI-DP-4λ formulation substantially improves continuity across subdomain interfaces by directly enforcing tangential-field and normal-flux traces.
- Normal-flux jumps fall from 9.1 × 10−7 with Robin TC to 2.2 × 10−12 with the proposed TC.This is a reduction of nearly six orders of magnitude at 10−6 stopping tolerance.
- Tangential-field jumps decrease from 1.4 × 10−8 with Robin TC to 1.2 × 10−10 with the proposed TC.The reduction exceeds two orders of magnitude at 10−6 stopping tolerance.
- Both interface continuities improve as the stopping tolerance is tightened, while the proposed TC maintains much smaller normal-flux jumps.
IV. CONCLUSION
The paper introduces a two-channel transmission condition and FETI-DP-4λ method for large-scale three-dimensional electromagnetic domain decomposition. Across diverse scenarios, the method reduces interface-equation iterations and interface jumps while preserving accuracy.
- The proposed method combines a Faraday/Ampère-Maxwell two-channel transmission condition with a mixed E-B FETI-DP-4λ formulation.
- Across varying geometries, partitionings, discretizations, subdomain configurations, and evanescent-wave content, iteration counts decrease by up to several-fold.
- Tangential-field and normal-flux interface jumps decrease by orders of magnitude while solution accuracy is preserved.
- The resulting solver supports robust, accurate, massively parallel, and scalable iterative domain decomposition for large-scale three-dimensional electromagnetic problems.