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Polynomial preserving recoveries of edge element method on Cartesian grids for the time-harmonic Maxwell equations with large wave number

Shuaishuai Lu, Haijun Wu

arXiv:2609.11064v1math.NAcs.NA

TL;DR

At large wave numbers, Maxwell edge-element approximations face pollution in addition to interpolation error, and prior recovery theory had not treated its role explicitly. This paper develops recovery operators for the curl and electric field and a CIP-EEM with an added normal-jump penalty. Recovery yields superconvergence but does not remove pollution, while suitable CIP-EEM penalties improve phase error by two orders in κh.

  • Problem

    For Maxwell edge elements, prior superconvergence work lacked wave-number-explicit analysis of whether recovery remains effective when pollution is present at large wave numbers.

    Source: §1 · p. 2

  • Method

    The paper develops polynomial-preserving recovery operators for the curl and electric field, and a CIP-EEM with an additional normal-component jump penalty.

    Source: §1 · pp. 2–3

  • Results

    With suitable penalty parameters, CIP-EEM improves phase error by two orders in κh; PPR improves interpolation error but does not reduce intrinsic pollution.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Under κ^3h^2Csol ≤ C0, recovery gives second-order superconvergence, while reducing pollution requires a discretization change such as the proposed CIP-EEM.

    Source: §1 · p. 3

    PPR does not remove the pollution term, and the recovered-curl estimator is reliable only when κ^3h^2Csol ≲ κh.

    Source: §1 · p. 3, §3.4 · pp. 20–21

Abstract

from arXiv · show

This paper considers the lowest-order first type N\'{e}d\'{e}lec edge element method (EEM) on Cartesian grids for the three-dimensional time-harmonic Maxwell equations with a large wave number. New polynomial preserving recovery (PPR) operators are proposed for the curl of the edge element solution and for the solution itself, respectively. Under the condition that $\kappa^3 h^2 C_{\mathrm{sol}}$ is sufficiently small, second-order superconvergence estimates are proved for both the recovered curl and the recovered solution, where $\kappa$ is the wave number, $h$ is the mesh size, and $C_{\mathrm{sol}}$ is a stability constant associated with the Maxwell solution operator. In particular, the analysis shows that the proposed PPR procedures cannot mitigate the well-known pollution effect inherent to the EEM. To reduce the pollution error, we further propose a new continuous interior penalty edge element method (CIP-EEM) that incorporates an additional normal-jump penalty term. It is shown that by appropriately choosing the penalty parameters, the new CIP-EEM can improve the phase error by two orders in $\kappa h$. Numerical experiments are presented to confirm the theoretical superconvergence results and to demonstrate that the CIP-EEM can effectively reduce the pollution error in the high-frequency regime.

1. Introduction

This paper studies whether polynomial-preserving recovery remains effective for high-frequency Maxwell edge-element solutions, where pollution accompanies interpolation error. It proves recovery estimates for the curl and electric field, shows recovery does not remove pollution, and proposes a CIP-EEM variant to reduce phase error.

1. Introduction

  • For large wave numbers, the EEM error includes interpolation error O(κh) and pollution error O(κ^3h^2Csol), with Csol the Maxwell solution operator’s stability constant.Source: §1 · p. 2
  • The curl-recovery analysis targets a gap in prior edge-element theory: wave-number-explicit analysis and the role of pollution at large wave numbers had not been addressed.The Maxwell analysis is more delicate than a routine Helmholtz adaptation because it is vector-valued and curl has a large nontrivial kernel.Source: §1 · p. 2
  • Under κ^3h^2Csol ≤ C0, curl recovery improves the interpolation component from O(κh) to O(κ^2h^2), while the pollution term remains the same order.C0 is independent of κ and h.Source: §1 · p. 3
  • In the high-frequency regime, PPR cannot remove intrinsic pollution, and the recovered-curl error estimator is reliable only when κ^3h^2Csol ≲ κh.The analysis asks whether recovery remains effective in the preasymptotic regime where pollution is present.Source: §1 · pp. 2–3
  • The paper constructs a function-value recovery operator and proves superconvergence estimates for the electric field, addressing the lowest-order element’s generally non-second-order L2 error.Source: §1 · p. 3
  • With suitable penalty parameters, the new CIP-EEM improves phase error by two orders in κh.It adds a penalty on jumps of the solution’s normal components; numerical results indicate further accuracy gains with recovery and reductions in interpolation and pollution errors.Source: §1 · p. 3

2. Preliminaries

The preliminaries establish the paper’s notation and function spaces, then state the Maxwell variational problem and assumptions used for stability and discretization. They also specify the Cartesian-grid setting and lowest-order edge-element method used in the subsequent analysis.

2.1. Spaces, norms, and notations.

  • The paper defines its Sobolev-space notation, L2 inner products, Maxwell energy norm, and conventions for comparing quantities up to constants independent of h and κ.Boldface denotes vector-valued spaces, and when D = Ω the paper abbreviates the associated norms.Source: §2.1 · p. 4

2.2. Variational formulation and EEM.

  • The Maxwell variational problem has a unique weak solution when κ² is not a positive Maxwell eigenvalue.The preliminaries assume κ ≳ 1 and κ² is not an eigenvalue.Source: §2.2 · pp. 4–5
  • For the variational problem, the forcing is decomposed into orthogonal divergence-free and gradient components, supporting the solution and uniqueness argument.Source: §2.2 · pp. 5–6
  • The superconvergence analysis assumes a rectangular box partitioned into uniform, shape-regular cuboidal elements and uses the lowest-order first-type Nédélec space.Source: §2.2 · p. 6

2.3. Interpolation estimates.

This section sets out interpolation and Helmholtz tools, then establishes stability and regularity properties for auxiliary Maxwell problems. These results provide analytic foundations for the later error estimates.

2.3. Interpolation estimates.

  • The lowest-order edge-element interpolant is defined by matching edge moments, and it maps V into Vh.Interpolation estimates are stated, with their proof attributed to [11, Theorem 10.2].Source: §2.3 · p. 6

2.4. Helmholtz decomposition and embedding results. \b

  • Under suitable structural conditions, smoothness of a vector field’s curl yields improved smoothness of the field.For the stated spaces on a convex polyhedron, fields belong to H1(Ω).Source: §2.4 · p. 7

2.5. Auxiliary Maxwell problems.

  • Two auxiliary Maxwell problems are introduced; each has a unique solution with stability estimates, and convexity plus divergence-free data supports additional regularity estimates.The problems differ in their equations, including a term involving P⊥ in one formulation.Source: §2.5 · pp. 7, 9

2.6. Elliptic projections.

The section establishes quasi-optimality for two elliptic projections and develops L2 superconvergence estimates for divergence-free error components.

2.6. Elliptic projections.

  • Ph is quasi-optimal when κh is sufficiently small, while ˜Ph is well-defined and quasi-optimal for arbitrary h > 0.Both estimates compare projection error with the best approximation from Vh.Source: §2.6 · p. 10
  • The projection estimates rely on the condition κh ≤ ˜C0 for Ph, whereas ˜Ph has no such mesh restriction in the stated result.The proof invokes coercivity and Galerkin orthogonality.Source: §2.6 · p. 10
  • The analysis establishes L2 superconvergence estimates for divergence-free components of both the elliptic projection error and the EEM error.The section also gives an estimate for the divergence-free part of E − Eh.Source: §2.6 · p. 10, §2.7 · p. 11

2.7. Preasymptotic error estimates for EEM.

The section gives preasymptotic error bounds for EEM under a mesh condition involving the wave number and solution stability constant. Its analysis also uses interpolation superconvergence on cuboid meshes.

2.7. Preasymptotic error estimates for EEM.

  • When κ^3h^2Csol ≤ C0, the EEM error is bounded by (κh + κ^3h^2Csol)(κ^-1∥curl E∥1 + |E|1).The theorem also bounds the error by (1 + κ^2hCsol) times the best-approximation error.Source: §2.7 · p. 11
  • The error analysis relates the energy-norm error to approximation error and the L2 error, then uses stability estimates to bound it.The proof obtains a relation of the form ∥E − Eh∥ ≲ ∥E − vh∥ + κ∥E − Eh∥.Source: §2.7 · p. 12
  • The analysis recalls an interpolation superconvergence property for E and its edge-element interpolant on cuboid meshes [32, Lemma 3].The property is used as a preliminary result for the error estimates.Source: §2.7 · p. 11, §2.8 · p. 12

2.8. Some supercloseness results.

The section establishes supercloseness between the EEM solution, elliptic projections, and the edge-element interpolant under stated mesh conditions. It also bounds an intermediate error component in terms of the solution regularity and stability constant.

2.8. Some supercloseness results.

  • If κh ≤ ˜C0, the elliptic projection PhE is superclose to the edge-element interpolant ˆEh.This is stated as a separate supercloseness result.Source: §2.8 · pp. 12–13
  • If κ^3h^2Csol ≤ C0, the EEM solution Eh is superclose to the edge-element interpolant ˆEh.The theorem’s estimate is stated under this mesh condition.Source: §2.8 · p. 14
  • The intermediate component θh = Eh − PhE is bounded by κ^2h^2Csol(∥E∥2 + κ∥E∥1) under the stated mesh condition.The bound is used with Lemma 2.12 to obtain the supercloseness estimate.Source: §2.8 · p. 14

3. The PPR curl recovery

The section extends polynomial preserving recovery to the curl of lowest-order Nédélec edge element solutions on three-dimensional cuboid meshes. It defines a local least-squares recovery and establishes its properties and superconvergence behavior, while showing that recovery does not remove pollution error.

3.1. Construction of the curl recovery operator Gh.

  • The recovered curl is constructed elementwise by fitting a vector-valued polynomial to edge data on an eight-element patch, then taking its curl on the target element.Repeating the construction over all elements gives the global recovered field.Source: §3.1 · pp. 14–15, §3 · p. 14

3.2.2. Tensor-product bases indexing.

3.2. Well-posedness of problem (3.5).

  • The local least-squares systems are uniquely solvable: their coefficient matrices scale with the cuboid side lengths and the reference matrix has rank 18.This establishes that the curl recovery operator is well-defined.Source: §3.2.3 · p. 17

3.3. Properties of Gh.

  • Each recovered-curl component is an affine combination of curl values at fixed sampling points, with coefficient functions that are independent of patch scaling.The operator is also shown to be stable and polynomial preserving.Source: §3.3 · p. 18

3.4. Error estimates for curl recovery.

  • Under κ^3h^2Csol ≤ C0, curl recovery improves the interpolation error from O(κh) to O(κ^2h^2), but leaves the pollution error unchanged.The analysis and numerical tests indicate that the unchanged pollution term is not merely an overestimate.Source: §3.4 · p. 20
  • When the pollution error is significant, the recovery correction is not a reliable estimate of the curl error.The correction is instead asymptotically exact for the error between the exact curl and the elliptic projection’s curl.Source: §3.4 · p. 21

4. The PPR function value recovery

This section introduces a companion local recovery for the vector field itself, using least-squares fitting on patches. Under the stated mesh condition, the recovered field has second-order accuracy in L2, although its interpolation improvement does not remove pollution error.

4.1. Construction of the function value recovery operator Fh.

  • Fh reconstructs a higher-order vector field from edge data by local least-squares fitting on patches, and its elementwise curl gives the recovered curl Gh.The procedure is repeated over the mesh to form the global recovered field.Source: §4.1 · p. 22

4.2. Superconvergence estimates for the function value recovery operator.

  • The section establishes superconvergence estimates for Fh using its L2 stability and polynomial-preserving property alongside interpolation estimates.The estimates parallel the curl-recovery analysis.Source: §4.1 · p. 22
  • The recovered field is second-order accurate in L2, but its interpolation improvement leaves the pollution component of the error in place.The superconvergence estimates are established under κ^3h^2Csol ≤ C0.Source: §4.2 · p. 23

5. CIP-EEM and Numerical Results

The section introduces a CIP-EEM with an additional normal-jump penalty and uses dispersion analysis to select penalty parameters. Numerical experiments show that it reduces high-frequency pollution, while PPR alone does not prevent the standard EEM’s high-frequency errors.

5.1. Continuous interior penalty edge element method (CIP-EEM).

  • The CIP-EEM adds a normal-component jump penalty to the classical tangential-curl jump penalties, and complex penalty parameters provide flexibility for dispersion control and stability.The additional J2 term is motivated by estimators involving jumps of the normal component of discrete solutions.Source: §5.1 · p. 25

5.2. Dispersion analysis of the CIP-EEM.

  • Dispersion analysis on a uniform Cartesian mesh selects γt = γn = −1/12 to reduce the phase difference between exact and discrete wave numbers.The analysis studies discrete Bloch waves and relates phase difference to pollution error.Source: §5.2 · pp. 25, 27

5.3. Numerical experiments.

  • For κ = 5, PPR reconstruction errors for EEM converge with slope −2 in both the H(curl) seminorm and L2 norm.EEM and CIP-EEM behave essentially identically in this low-frequency regime, where pollution is negligible.Source: Figure 2 · p. 28, §5.3 · pp. 28–29
  • At κ = 50, standard EEM errors remain at or above 100% on fine meshes, whereas CIP-EEM stays stable and its reconstructed quantities converge at rates exceeding first order in both norms.The paper attributes the improvement to appropriately chosen complex penalty parameters.Source: Figure 3 · p. 29, §5.3 · pp. 29–30
  • Under κh0 = 1, CIP-EEM errors remain uniformly small, below approximately 0.3 even at κ = 90, while EEM errors deteriorate notably for κ > 10.The EEM PPR reconstructions do not improve accuracy as pollution grows.Source: Figure 4 · p. 30, §5.3 · p. 30

Appendix A.

Appendix A identifies A0 as an 18 by 18 matrix.

Appendix A.

A.1. Explicit form of A0. 

The appendix states that A0 is invertible.

A.1. Explicit form of A0.

A.2. Proof of Lemma 3.1.

The proof reduces Lemma 3.1 to the reference patch, constructs a finite-sample affine representation there, and transfers it to each physical patch.

A.2. Proof of Lemma 3.1.

  • A diagonal affine map and the covariant Piola transform transfer the reference-patch construction to physical patches while preserving the edge moments.The curl transforms componentwise under the diagonal map, supplying the scaling used in the transfer.Source: §A.2 · pp. 31–32
  • On the reference patch, each component of d curl of the least-squares fit is an affine combination of corresponding sampled components from the edge-element field.The coefficients depend on patch geometry and sum to one.Source: §A.2 · pp. 32, 35, 36
  • The proof establishes the affine-combination property (3.9) on physical patches with 12 sampling points per component.The final transfer yields the desired property with M_i = 12 for i = 1, 2, 3.Source: §A.2 · pp. 32, 36
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