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Finite Element Analysis of Maxwell's Equations with Time-Varying Coefficients and Nonlinear E-H Coupling: Convergence and Exponential Decay
Jan Renner, Irwin Yousept
TL;DR
The paper studies finite element analysis for a non-autonomous Maxwell system combining time-varying material parameters with nonlinear E-H coupling, a setting where standard convergence tools do not apply. It develops a Cauchy-type convergence argument and a sup-max decay analysis, establishing uniform convergence on finite intervals and conditional fully discrete unconditional exponential decay on the infinite horizon.
Problem
Finite element approximations for a non-autonomous Maxwell system with simultaneous nonlinear distributed and boundary E-H coupling had not yet been analyzed, and standard monotonicity and discrete compactness approaches fail in this setting.
Source: §1 · pp. 1–2Method
The paper uses a Cauchy-type argument in a time-dependent Hilbert space for convergence and a nonlinear sup-max problem under positive homogeneity for decay analysis.
Source: §1 · p. 2Results
Under sufficient condition (B5), the analysis yields fully discrete unconditional exponential decay; without (B5), a counterexample shows exponential energy decay fails.
Source: §1 · p. 2Takeaways & Limitations
The paper establishes convergence and decay results for this non-autonomous nonlinear Maxwell setting despite the inapplicability of standard convergence tools.
Source: §1 · p. 2Even with strengthened assumptions, monotonicity and discrete compactness arguments do not suffice to establish the needed limit in (48).
Source: §3.4 · p. 11
Abstract
from arXiv · showhide
This paper analyzes finite element approximations for Maxwell's equations with time-dependent coefficients and nonlinear E-H coupling in both the total current density and the Silver-M\"uller-type boundary condition. The nonlinear coupling and the time-varying material parameters prevent the use of standard analytical tools, including the discrete compactness property and the Minty-Browder argument, making the numerical analysis particularly challenging. As the main novelty, we establish two primary contributions: Uniform convergence on every finite time interval $[0,T]$ and exponential stability on the infinite time horizon $[0,\infty)$. The uniform convergence is established via a Cauchy-type argument within a time-dependent Hilbert space framework that entirely bypasses Minty's trick and the discrete compactness argument. Furthermore, under a positive homogeneity assumption on the nonlinearities, the exponential stability is analyzed by means of a nonlinear sup-max problem arising from ratios of consecutive discrete energies. With this strategy, we ultimately prove a fully discrete unconditional exponential decay result under a specific sufficient condition. In the absence of this condition, a concrete counterexample shows that the energy decay fails to hold. Numerical experiments are provided to validate the theoretical findings.
1. Introduction.
The paper analyzes finite element approximations for a non-autonomous Maxwell system with nonlinear E-H coupling, addressing a setting not previously analyzed with both distributed and boundary coupling. It establishes uniform convergence on finite time intervals and fully discrete unconditional exponential decay on the infinite time horizon.
1. Introduction.
- The problem combines time-varying material parameters with nonlinear E-H coupling in the total current density and boundary condition.The paper notes that varying temperature and evolving microstructure can influence material properties, while magnetic-field-dependent current arises in settings such as type-II superconductivity.Source: §1 · pp. 1–2
- The paper establishes uniform convergence on [0, T] and fully discrete unconditional exponential decay on [0, ∞) for the considered Maxwell system.The decay result uses sufficient condition (B5); a counterexample shows exponential energy decay fails in its absence.Source: §1 · p. 2
- A Cauchy-type argument in a time-dependent Hilbert space framework proves uniform convergence without Minty-Browder or discrete compactness arguments.Time-varying material parameters and H-dependence in the total current density make the system and its fully discrete scheme generally non-monotone.Source: §1 · p. 2
- The exponential-decay analysis assumes positive homogeneity of b and J_tot and uses a nonlinear sup-max problem.Condition (B5), together with this analysis, yields the decay result; without (B5), the paper gives a counterexample to exponential energy decay.Source: §1 · p. 2
2. Preliminaries.
The preliminaries set notation and introduce a Hilbert-space framework for fields with tangential traces. They also state regularity and structural assumptions on the material parameters, source, current density, and boundary nonlinearity.
2. Preliminaries.
- Space-time fields u are also treated as time-indexed Banach-valued mappings t ↦ u(·, t).The section sets notation for norms, dual spaces, duality pairings, and Hilbert-space scalar products.Source: §2 · p. 3
- The analysis uses V, a space of H(curl) fields with square-integrable tangential traces satisfying a boundary constraint, and a partial integration formula for fields in V.The tangential trace operator is linear and bounded; V is introduced as the main Hilbert space for the analysis.Source: §2 · p. 3
- The time-dependent permittivity and permeability satisfy uniform lower and upper bounds on their quadratic forms.The external current source is assumed to belong to H1(0, T; L2(Ω)).Source: §2 · p. 3
- The boundary nonlinearity b is monotone, vanishes at zero, and satisfies the stated coercivity and growth conditions.Assumption (A6) is required for stability but not for well-posedness.Source: §2 · p. 4
3. Finite element analysis.
This section develops the mixed finite-element framework and analyzes its fully discrete scheme, establishing well-posedness, mesh-uniform stability, and convergence to the unique solution on finite time intervals. It also studies exponential stability over an infinite time horizon, proving unconditional exponential decay under stated assumptions and illustrating that decay can fail when a sufficient condition is not met.
3.1. Finite element spaces.
- Under Assumption 2.1 on a quasi-uniform triangulation family, the method uses first-family Nédélec and piecewise-constant finite element spaces, with Π_hu→u in V for all u∈V as h→0.Source: §3.1 · p. 4
3.2. Fully discrete scheme.
- Under a CFL-type condition, the fully discrete scheme has a unique solution; the proof condition arises from the inverse curl estimate and is not optimal.The condition also reflects the time dependence of the magnetic permeability.Source: §3.2 · pp. 5–6
- Under (A1)–(A5), uniformly positive semidefinite ∂tµ, and the stated growth and coefficient conditions, the scheme admits a solution for every h > 0 and N ∈N; inequality (26) gives at most one solution.Source: §3.2 · p. 7
- Although the coupling is monotone separately in each variable, its induced nonlinearity is not monotone, so the Minty–Browder argument generally cannot be applied.The cut-off example satisfies the proposition’s growth condition despite this failure of monotonicity.Source: §3.2 · p. 8
3.3. Stability.
- Projected initial data converge in V×L2(Ω), and the construction makes the discrete scheme hold at n=0, a property used in the stability proof.The initial discrete increment is also required to be bounded in L2(Ω).Source: §3.3 · p. 8
- The discrete solutions satisfy a stability bound with constants independent of mesh size and time-step count once the stated threshold on N is met.The stability proposition requires N > max{β, M(h)}; a bound for h > 0 is also stated.Source: §3.3 · p. 9
3.4. Convergence.
- A modified Cauchy argument in a differentiable, time-dependent Hilbert space establishes convergence without relying on Minty’s trick or discrete compactness.The standard limit-passage approach fails even under strengthened assumptions, motivating the time-dependent inner product and the limit-inferior strategy.Source: §3.4 · pp. 11–12
- The fields converge uniformly in C([0, T], L2(Ω) × L2(Ω)) to the unique solution of (P) as h → 0.The limit belongs to W 1,∞(0, T; L2(Ω) × L2(Ω)) ∩ L∞(0, T; V × L2(Ω)).Source: §3.4 · p. 12
3.5. Exponential stability.
- The assumptions set µ(t)=µ∞ for t≥Tµ≥0 with µ∞ uniformly positive definite, require b(x, λξ)=λb(x, ξ) for a.e. boundary x, all ξ∈R3 and λ≥0, and set f(t)=0 for t≥Tf≥0.They also state Jtot(x, ξ, η, t)=Jtot(x, ξ, η′, t).Source: §3.5 · p. 15
- Theorem 3.13 establishes fully discrete unconditional exponential stability under Assumptions 3.1 and 3.2.The section assumes the discrete scheme is well-posed for every time step.Source: §3.5 · pp. 16–17
- The section’s exponential-decay corollary includes the case T_µ = 0 when the initial discrete state belongs to W_h.In that case, the proof uses the second equation of the discrete scheme to show the states remain in W_h.Source: §3.5 · p. 17
- With J_tot = 0, the example shows that the energy is not exponentially decaying and condition (B5) fails.For J_tot = 2E, (B5) holds when τ ≤ 1/2, and the energy decays exponentially.Source: §3.5 · p. 20
4. Numerical results.
The numerical section reports two tests: one indicates first-order convergence, and the other confirms exponential energy decay consistent with the analysis. The computations use the fully discrete finite element scheme and a Newton-based iterative solver.
4. Numerical results.
- The scheme is computed using a fixed-point iterative algorithm with Newton’s method in Legacy FEniCS (2019.1.0).The numerical section presents one convergence test and one exponential-stability test.Source: §4 · pp. 20–21
4.1. Convergence test.
- The convergence test measures errors against an exact solution over T = 1.75 while varying the spatial and temporal discretizations.The reported experimental order of convergence compares errors from consecutive tests with N < N′ and h > h′.Source: §4.1 · p. 21
- The first test indicates a first-order convergence rate, with the experimental order approaching 1 as the mesh is refined and the time resolution increased.The reported behavior agrees with Theorem 3.8.Source: §4.1 · p. 21, Table 1 · p. 21, Figure 1 · p. 22
4.2. Exponential decay test.
- For the cylindrical pipe coil Ωp, with Assumption 3.1 satisfied at tm = 0.2, numerical results confirm exponential decay behavior of the discrete electromagnetic energy, consistent with Theorem 3.13.Figure 2 shows the discrete electromagnetic energy for tn ≥ 0.002 and its exponential least-squares fit over [0.2, 12] on a logarithmic scale.Source: §4.2 · p. 21, Figure 2 · p. 22
A. Appendix A.
The appendix proves uniqueness for solutions of the continuous problem by applying Grönwall’s inequality to the difference of two solutions.
A. Appendix A.
- The uniqueness argument begins by subtracting the two solution pairs to define their electric- and magnetic-field differences.Source: A. Appendix A · p. 23
- The difference estimate includes the changes in total current density and in the nonlinear boundary term evaluated against the corresponding field differences.Source: A. Appendix A · p. 23
- Grönwall’s inequality yields equality of the two solutions once the difference estimate is established.The estimate uses uniform positive definiteness of ε and μ and monotonicity in the total-current and boundary terms.Source: A. Appendix A · p. 23
B. Appendix B.
The appendix uses the Nédélec interpolation operator to control approximation error in the H(curl) norm and derive an estimate for the Hilbert projection.
B. Appendix B.
- For 1/2 < s ≤ 1, the Nédélec interpolant satisfies an H(curl) error bound proportional to h^s times the H^s(curl) norm.The bound’s constant is independent of h and u.Source: B. Appendix B · p. 23
- The trace of u is well-defined in L2(∂Ω) when u belongs to H^s(curl) with s > 1/2.Source: B. Appendix B · p. 23
- The interpolation estimate and density of smooth functions in V yield the stated estimate for the Hilbert projection.Source: B. Appendix B · p. 23
C. Appendix C.
The appendix develops estimates for the fully discrete scheme, using discrete Grönwall arguments to obtain bounds uniform in the time and spatial discretizations.
C. Appendix C.
- Lemma C.1 is applied within the estimate, and its proof uses an index shift to complete the stated identity.Source: C. Appendix C · pp. 23–24
- The proof estimates successive discrete field differences by subtracting scheme equations at adjacent time indices and testing with those differences.Source: C. Appendix C · p. 24
- The estimates also invoke the assumptions, Young’s inequality, and Cauchy–Schwarz to bound terms in the discrete energy argument.Source: C. Appendix C · pp. 24–26
- Discrete Grönwall yields uniform bounds for the discrete electric and magnetic fields after the estimates for terms I–III are collected.The derivation requires N > β to absorb the term corresponding to k = m.Source: C. Appendix C · pp. 26–27