Eigen-Decomposition Test for Maxwell Solvers

• Eigen-Decomposition-Based Test is a computational method that uses spectral diagonalization to efficiently solve Maxwell's equations while preserving physical constraints.
• It decomposes discrete vector fields into divergence-free and curl-free subspaces, eliminating spurious modes and ensuring exact compliance with Gauss's law.
• The method features a fast, matrix-free solver with O(n² log n) complexity that outperforms traditional LU and multigrid solvers in terms of speed and memory usage.

An eigen-decomposition-based test is a methodology in computational mathematics and scientific computing that systematically exploits the explicit eigen-decomposition of discrete operators—most prominently in the context of partial differential equations such as Maxwell's equations—to construct efficient, robust, and structure-preserving solvers. In the context of Maxwell's equations on rectangular domains, eigen-decomposition-based tests leverage exact spectral diagonalization (often using trigonometric transforms) to provide fast solvers that guarantee divergence-free constraints and Gauss's law compliance at the discrete level, eliminating spurious modes and outperforming conventional approaches in efficiency and memory usage (Wang et al., 13 Mar 2025).

1. Discrete Eigen-Solutions and Transform Diagonalization

The core of the approach is the explicit eigen-decomposition of the discrete operators resulting from the spatial discretization of Maxwell's equations with edge finite elements. In a rectangular domain with lowest-order elements, the discrete curl-curl operator and associated gradient/divergence matrices are constructed from tensor products of one-dimensional second-difference and difference operators.

These matrices are simultaneously diagonalized by discrete sine and cosine transform matrices (denoted S and C), yielding

• $S^{-1} A S = \Lambda^2$, with $\Lambda$ diagonal and entries $\tau_i = -2 \sin(i\pi/(2n))$,
• $S^{-1} D S = \Sigma$, with $\Sigma$ diagonal and entries $\sigma_i = 2\left(2 + \cos(i\pi/n)\right)$.

The discrete eigenvalue problem can thus be decomposed into independent modes. Two families of discrete eigen-solutions are obtained:

Eigenvalue Family	Characterization	Eigenstructure
Nonzero eigenvalues	Divergence-free eigenfunctions (physical modes)	Explicit formulas using tensor products of S/C vectors
Zero eigenvalue	Curl-free eigenfunctions (gradient fields)	Characterized by vanishing of the discrete curl operator

This explicit diagonalization significantly reduces the algebraic complexity of the global system, making spectral techniques natural and highly effective.

2. Discrete Hodge–Helmholtz Decomposition Framework

The method induces a discrete Hodge–Helmholtz (or Helmholtz) decomposition at the algebraic level. The space of discrete vector fields (electric fields) decomposes as a direct sum:

• Divergence-free (physical) subspace: spanned by nonzero eigenvalue eigenfunctions,
• Gradient (curl-free) subspace: spanned by zero eigenvalue eigenfunctions.

The eigen-decomposition ensures that physical constraints—such as divergence-freeness or compliance with Gauss's law—hold exactly for the corresponding subspaces in the discrete system. For instance, the divergence-free constraint appears as an explicit equation in the transformed coordinates, e.g., $$\Gamma \widehat{U} \Sigma + \Sigma \widehat{V} \Gamma^T = 0$$, directly enforcing the physical law.

3. Fast Solver Architecture

The spectral diagonalization leads directly to a fast solver architecture:

• The right-hand side (sources) and solution unknowns are transformed into the spectral basis via fast sine and cosine transforms (FST/FCT).
• The system reduces to a block-diagonal structure, decoupling the modes.
• Each modal problem is scalar (or very low-dimensional) and can be solved independently by explicit formulas, such as:

$$\widehat{U}(i+1, j) = \frac{h^2 \widehat{F}(i+1, j)}{(\lambda^{(h)}_{i,j} + \alpha)(h^2 \sigma_j / 6)}$$

• The solution is recovered in physical space by inverse transforms.

Enforcement of divergence-free or Gauss's law conditions is guaranteed by the basis construction and the structure of the solution expansion.

4. Computational Complexity and Efficiency

Because the main computational cost lies in forward and inverse FST/FCT operations and diagonal scaling, the total complexity for a two-dimensional $n \times n$ grid is $\mathcal{O}(n^2 \log n)$. This is on par with the best available Poisson/Helmholtz solvers based on trigonometric transforms.

Key efficiency characteristics:

Solver Type	Asymptotic Cost	Storage Complexity	Physical Constraint
Eigen-based (this)	$\mathcal{O}(n^2\log n)$	Minimal (matrix-free)	Exact (discrete)
LU or Direct	$\mathcal{O}(n^6)$	Dense/factorization	May require Lagrange mult.
Multigrid/Iterative	Problem-dependent	Varies (often sparse)	Preconditioning needed

Moreover, the method is matrix-free, drastically reducing memory consumption (often by more than two orders of magnitude versus direct solvers), and is well-suited for preconditioning variable-coefficient problems.

5. Comparison to Conventional Solvers and Stability

Conventional methods—direct LU or iterative multigrid—typically solve the saddle-point systems induced by the mixed formulation, with attendant large fill-in, higher computational cost, and risk of spurious nullspace modes. The eigen-decomposition-based test, by contrast:

• Eliminates spurious solutions (nonphysical, zero-eigenvalue modes) by strict enforcement of the discrete Helmholtz–Hodge decomposition,
• Achieves first-order convergence in $L^2$ norm and $H(\mathrm{curl})$ semi-norm, matching theoretical predictions,
• Outperforms LU and multigrid solvers in speed, robustness, and storage, as evidenced by benchmark tables: for large $n$, LU solvers may fail due to memory, while the eigen-based solver completes in milliseconds to seconds.

6. Preconditioning and Extensions

The method is also utilized for efficient preconditioning of variable-coefficient Maxwell problems. The constant-coefficient eigen-decomposition-based solver is employed as a preconditioner in CG-type methods, dramatically reducing the iteration count and preserving $\mathcal{O}(n^2\log n)$ scaling. The approach is extensible:

• Higher-order spectral elements (via the same diagonalization architecture),
• Mixed boundary conditions,
• Gauss's law-preserving extensions using Lagrange multipliers,
• Potential for three-dimensional and nonrectangular domains.

7. Numerical Results and Impact

Numerical tests confirm:

• Rapid, robust convergence with low memory consumption,
• Correct enforcement of physical constraints,
• Orders-of-magnitude speedup compared to traditional solvers.

Furthermore, the eigen-decomposition-based framework provides not just an efficient numerical method, but also a theoretical guarantee of structure preservation (no spurious modes, exact discrete constraints), positioning it as a foundational approach for modern high-performance computation of electromagnetic problems (Wang et al., 13 Mar 2025).

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This methodology thus constitutes a paradigmatic instance of how explicit operator eigen-decomposition, in conjunction with spectral transforms, yields fast, robust, and physically faithful solvers for large-scale PDE-based simulation.