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Finite Element Methods for Maxwell's Equations
Peter Monk, Yangwen Zhang
TL;DR
The paper addresses how finite element methods approximate time-harmonic Maxwell equations with spatially varying coefficients, especially where DG error analysis remains less developed. It surveys conforming edge elements and three DG families, finding that DG methods offer flexibility and other practical advantages despite weaker theoretical justification, particularly for inhomogeneous media.
Problem
Error estimates are comparatively well established for conforming edge elements with piecewise smooth coefficients but remain less advanced for DG methods in spatially varying media.
Method
The paper surveys conforming edge elements and three DG families—Interior Penalty, Hybridizable DG, and Trefftz methods—for time-harmonic Maxwell equations.
Results
DG methods offer flexibility for local discretization, hanging nodes, polyhedral meshes, conservation, and hp-adaptivity, while conforming edge methods remain the best-understood theoretically.
Takeaways & Limitations
DG methods have significant potential advantages over conforming methods, including easier hp-adaptivity and reduced degrees-of-freedom costs through hybridizable or Trefftz variants.
Takeaways & Limitations
The survey considers only the simplest boundary value problem and notes that DG schemes are less theoretically justified, especially for inhomogeneous media.
Abstract
from arXiv · showhide
We survey finite element methods for approximating the time harmonic Maxwell equations. We concentrate on comparing error estimates for problems with spatially varying coefficients. For the conforming edge finite element methods, such estimates allow, at least, piecewise smooth coefficients. But for Discontinuous Galerkin (DG) methods, the state of the art of error analysis is less advanced (we consider three DG families of methods: Interior Penalty type, Hybridizable DG, and Trefftz type methods). Nevertheless, DG methods offer significant potential advantages compared to conforming methods.
1. Introduction
The paper surveys finite element approximations for time-harmonic Maxwell equations, emphasizing spatially varying materials and discontinuous Galerkin methods. It formulates the problem variationally and identifies divergence control, coefficient discontinuities, and non-coercivity as central analytical issues.
- Motivation: Maxwell’s equations model electromagnetic-wave propagation, motivating direct discretization for inhomogeneous or anisotropic materials.The survey focuses on finite element methods for frequency-domain electromagnetic waves, especially DG methods alongside conforming methods.
- Why DG methods: DG methods provide flexible local polynomial degrees, permit hanging nodes and general polyhedral elements, and support element-wise conservation.They are particularly attractive for hp-adaptivity compared with H(curl;Ω)-conforming edge elements.
- Model problem: The model assumes a bounded Lipschitz polyhedral domain, unit relative permeability, and piecewise W 1,∞ relative permittivity with strictly positive real part.The formulation also assumes nonnegative conductivity and divergence-free imposed currents F ∈L2(Ω).
- Variational formulation: The variational formulation is derived by multiplying the curl-curl equation by a conjugated test field and applying the curl integration-by-parts identity.The PEC boundary condition makes the boundary term vanish because the tangential test trace is zero.
- Analytical issues: The formulation lacks explicit divergence control, but testing with a gradient yields the constraint ∇⋅(ϵrE) = 0.This motivates analyzing the problem in a space incorporating the weighted divergence condition.
- Analytical issues: The sesquilinear form is not coercive, although a G˚arding inequality and compact embedding give uniqueness when κ is not a curl-curl eigenvalue.The paper assumes this restriction on κ thereafter.
2. Finite Elements and Maxwell’s Equations
The section contrasts continuous H1-based and conforming edge finite elements for Maxwell’s equations, emphasizing their approximation properties and limitations. It motivates edge elements through discrete divergence control and discusses challenges from reentrant geometry and discontinuous coefficients.
- 2.1. Stabilized H1 elements.: Continuous piecewise polynomial spaces can be constructed from three copies of standard scalar continuous piecewise k-degree finite element spaces.This construction is a major practical attraction, but continuity of both normal and tangential components is required.
- 2.1. Stabilized H1 elements.: On domains with reentrant corners or edges, stabilized continuous elements may converge to a function that does not solve Maxwell’s equations.In convex domains, convergence to the true solution occurs, whereas reentrant geometry creates an infinite-codimension approximation obstruction.
- 2.1. Stabilized H1 elements.: Discontinuous permittivity generally causes jumps in the normal electric-field component, so H1 elements require broken interface continuity and Nitsche transmission conditions.The resulting treatment is essentially the same as the Interior Penalty Discontinuous Galerkin method.
- 2.2. Edge Elements.: Nédélec edge elements use edge-, face-, and tetrahedron-associated degrees of freedom to define conforming H(curl;Ω) approximation spaces.The second family adds gradients of certain polynomials, while the first family is historically associated with Whitney and Nédélec elements.
- 2.2. Edge Elements.: Discrete compactness makes discrete divergence-free edge-element fields close to divergence-free functions, supporting convergence analysis for Maxwell source problems.The associated subsequence property yields an L2(Ω) limit satisfying ∇⋅(ϵ_ru)=0, with σ > 0 depending on Ω.
- 2.2. Edge Elements.: For piecewise constant ϵr with positive real part and κ away from Maxwell eigenvalues, sufficiently fine meshes admit a unique edge-element solution.This result provides a well-posedness guarantee for the conforming edge-element discretization under the stated coefficient and spectral assumptions.
3. Interior Penalty DG Methods
The IPDG method discretizes Maxwell’s equations with discontinuous vector polynomials, jump penalties, and a DG norm, with error analysis relying on regularity and duality arguments. For constant permittivity, the method has a well-posedness and error theorem; piecewise smooth coefficients require more sophisticated analysis.
- Variational construction: The method is formulated through elementwise integration by parts, jump and average traces, and a global DG identity over interior and boundary faces.The resulting identity is the basis for constructing the discretized IPDG method.
- Discretization: IPDG uses discontinuous vector polynomials and adds consistent symmetrization plus penalties on jumps across element faces.The penalty parameter α must be sufficiently large.
- Error measurement: The DG norm measures the solution error induced by the IPDG sesquilinear form.Its definition includes jump and curl contributions.
- Error analysis: For E ∈ H^s(Ω) and ∇×E ∈ H^s(Ω), s > 1/2, sufficiently large penalties yield a unique solution and an error estimate for sufficiently small h.The theorem assumes constant ϵr in Ω.
- Error analysis: The proof combines approximation estimates, discrete Helmholtz decomposition, an adjoint Maxwell problem, and a kickback argument.The adjoint argument exploits a divergence-free right-hand side to obtain additional smoothness.
- Coefficient dependence: Direct adjoint analysis is limited to constant ϵr, whereas piecewise smooth coefficients admit quasi-optimal estimates through a more sophisticated inf-sup analysis.This analysis is reported in [15, Section 6].
4. HDG Methods
HDG methods reduce the globally solved unknowns by hybridizing element interiors onto mesh-face traces, while retaining flexible choices of local spaces and fluxes. Their analyses include optimal convergence in selected settings, but some early formulations lacked error analysis or achieved only suboptimal rates.
- Hybridization: HDG eliminates element-volume degrees of freedom through static condensation, leaving a smaller global system involving only mesh-skeleton traces.The skeleton consists of all faces of the mesh.
- Hybridization: HDG was introduced to reduce the dimension of the global linear system and enable superconvergent recovery of variables of interest.Different local spaces and numerical fluxes define different HDG methods.
- Formulation: A standard HDG formulation introduces t = ∇×E and approximates t, E, and the tangential trace Ê using spaces Vh, Wh, and Mh.The local spaces are chosen within H1(K), H(curl;K), and L2(F), respectively.
- Formulation: The numerical flux includes a stabilization term with positive parameter τ, and one formulation uses polynomial spaces V(K) × W(K) × M(F) = [Pk(K)]^d × [Pk(K)]^d × [Pk(F)]^(d−1).The flux choice and spaces determine the particular HDG scheme.
- Existing analyses: Early HDG methods were shown well-posed, conservative, and consistent, but their study provided no error analysis and numerical experiments only in two dimensions.A three-dimensional domain-decomposition study also lacked convergence analysis.
- Existing analyses: M-decomposition analysis gives conditions for optimal convergence and superconvergence in two dimensions, while its extension to three dimensions remains challenging.Other reported analyses include suboptimal rates for some settings and optimal convergence under additional regularity assumptions.
5. The Ultra Weak Variational Formulation
The UWVF is a Trefftz DG method that uses elementwise Maxwell solutions, specifically plane waves, as trial and test functions. It supports scattering calculations and iterative solution, but its equivalence and convergence theory are restricted for complex coefficients, while conditioning deteriorates at high direction counts.
- Trefftz formulation: Trefftz DG methods approximate Maxwell fields with elementwise solutions of the underlying equations rather than piecewise polynomials.The UWVF is the particular Trefftz method developed for Maxwell’s equations.
- Analysis: For real, piecewise constant ϵr, UWVF is equivalent to an IPDG scheme and its convergence follows from plane-wave IPDG analysis; this equivalence fails for absorbing media.The theory requires the same plane-wave space for trial and test functions.
- Boundary formulation: The UWVF imposes a generalized boundary impedance condition with real λ > 0, parameter Q satisfying |Q| < 1, and boundary data g.Q = 1 recovers the PEC boundary condition considered earlier.
- Variational construction: Its derivation applies elementwise integration by parts and an isometry identity, then couples boundary impedance fluxes across element faces.The unknown flux is χK = ν × ∇×E|K + iκλ(E|K)T.
- Plane-wave basis: The discrete space uses pK independent propagation directions per element and two mutually orthogonal unit polarization vectors for each direction.The paper uses Hammersley points on the unit sphere for the directions in its example.
- Limitations: The discrete problem becomes rapidly ill-conditioned as p = maxK∈Th pk increases, and ad-hoc conditioning controls can limit accuracy.A combined Trefftz and finite-element basis is suggested as a possible response near singularities or on small elements.
- Numerical example: A sphere-scattering example uses 127,113 tetrahedra and 29–67 directions per element with a PML and absorbing outer boundary.The computed field is the real part of the scattered electric-field y component in the x−z plane.
6. Conclusions and the Future
Conforming edge finite elements have the strongest error-analysis foundation, while DG schemes remain less theoretically justified for inhomogeneous media. Future work includes tracking wave-number dependence and developing fast solvers for the discrete systems.
- Conforming edge finite elements are the best-understood method for error analysis and provide approximation results used to prove convergence of methods such as IPDG.
- The three discussed DG schemes are less theoretically justified, especially for inhomogeneous media.
- The surveyed error bounds do not explicitly track their dependence on the wave number κ.
- A central future challenge is obtaining fast solvers for the discrete matrix problem after discretization.
- UWVF results suggest that hybridized solvers operating on the mesh skeleton may help with wave-number dependence in iteration counts.