[Paper Review] Curl-free positive definite form of time-harmonic Maxwells equations well-suitable for iterative numerical solving

V.Е. Moiseenko, O. Ågren|arXiv (Cornell University)|Mar 28, 2020

Electromagnetic Simulation and Numerical Methods10 references2 citations

TL;DR

This paper proposes a curl-free, positive definite form of time-harmonic Maxwell’s equations using vector and scalar potentials (A, Φ), magnetic field (H), and electric displacement (D). By replacing curl operators with Laplacian and gradient-of-divergence terms and eliminating spatially varying coefficients, the method ensures diagonal dominance and positive definiteness in the discrete system, enabling efficient convergence with the conjugate gradient method. The approach achieves optimal iteration counts scaling as N^{1/3} in 3D problems.

ABSTRACT

A new form of time-harmonic Maxwells equations is developed and proposed for numerical modeling. It is written for the magnetic field strength, electric displacement, vector potential and the scalar potential. There are several attractive features of this form. The first one is that the differential operator acting on these quantities is positive. The second is absence of curl operators among the leading order differential operators. The Laplacian stands for the leading order operator in the equations for the magnetic field strength, vector potential and the scalar potential, while the gradient of divergence stands for the electric displacement. The third feature is absence of space varied coefficients in the leading order differential operators that provides diagonal domination of the resulting matrix of the discretized equations. A simple example is given to demonstrate the applicability of this new form of time-harmonic Maxwells equations.

Motivation & Objective

• To develop a form of time-harmonic Maxwell’s equations that avoids spurious modes introduced by curl operators in standard formulations.
• To eliminate sign indefiniteness in the resulting linear systems, enabling use of fast iterative solvers like conjugate gradients.
• To ensure diagonal dominance in the discrete system by removing spatially varying coefficients from leading-order differential operators.
• To simplify finite difference discretization by using the same mesh for all field quantities and facilitating boundary condition implementation.
• To demonstrate improved convergence behavior in iterative solvers through a 3D numerical benchmark.

Proposed method

• The formulation uses normalized vector potential A = a / (ik₀) and scalar potential Φ = -iφ / (k₀) to balance field magnitudes for consistent discretization.
• The equations are derived from the standard time-harmonic Maxwell’s equations using potentials, with the Coulomb gauge condition ∇·A = 0 applied.
• The system is cast into a quadratic form Q = ∫[|LA|² + |LH|² + |LD|² + |LΦ|²] dV, which is minimized to derive the weak form.
• The resulting weak form is discretized using standard 3D finite differences on a uniform structured grid, with volume-weighted node contributions.
• Boundary conditions are applied by setting Φ = 0 on the boundary and using consistent stencils for H, D, A, and Φ at boundaries and edges.
• The linear system Mx = b is solved using the conjugate gradient method without preconditioning, with convergence monitored via normalized residual and error.

Experimental results

Research questions

• RQ1 Can a curl-free, positive definite form of time-harmonic Maxwell’s equations be constructed using vector and scalar potentials?
• RQ2 Does the absence of curl operators and spatially varying coefficients in leading-order terms lead to diagonal dominance in the discrete system?
• RQ3 Can this formulation enable efficient convergence with the conjugate gradient method, avoiding stagnation common in sign-indefinite systems?
• RQ4 How does the iteration count scale with mesh size in 3D problems under this formulation?
• RQ5 Can a consistent, stable, and accurate finite difference discretization be implemented with a single mesh for all field components?

Key findings

• The proposed formulation results in a positive definite discrete system, enabling the use of the conjugate gradient method with guaranteed convergence.
• The leading-order differential operators are the Laplacian (for H, A, Φ) and the gradient of divergence (for D), eliminating curl operators from dominant terms.
• The absence of spatially varying coefficients in the leading-order operators ensures diagonal dominance in the resulting linear system matrix.
• Numerical results show smooth and rapid convergence of the conjugate gradient solver, with residual decreasing steadily over iterations.
• The optimal number of iterations to achieve half the discretization error scales approximately as N^{1/3} with total mesh node count N, indicating efficient iteration performance.
• The analytical solution for a 3D current source with sinusoidal components was successfully matched by the numerical solution, validating accuracy and consistency.