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Operator preconditioning of the Costabel FEM-BEM coupling for the time-harmonic Maxwell transmission problem

Quang Huy Nguyen, Van Chien Le, Kristof Cools

arXiv:2610.09765v2math.NAcs.NA

TL;DR

For time-harmonic Maxwell transmission with a Lipschitz interface, this paper reformulates the Costabel FEM-BEM coupling using a trace lifting operator and the Steklov–Poincaré operator. It proves equivalence with the original coupling under a resonance exclusion and constructs a block-diagonal preconditioner with a uniformly bounded spectral condition number.

  • Problem

    FEM-BEM systems combine sparse volume and dense boundary blocks, making direct solvers impractical, while existing preconditioning theory has limits for non-smooth surfaces and electromagnetic problems.

    Source: §1 · p. 2

  • Method

    The paper uses a trace lifting operator and the Steklov–Poincaré operator to reformulate the coupling on trace spaces, then constructs a block-diagonal preconditioner within the operator preconditioning framework.

    Source: §1 · p. 2, §6 · p. 14

  • Results

    Outside the resonance sets, the reformulated coupling is equivalent to the Costabel coupling, and the preconditioned matrix has a uniformly bounded spectral condition number.

    Source: §1 · p. 3, §6 · p. 14

  • Takeaways & Limitations

    The resulting preconditioner provides an efficient method for the discrete system arising from the Costabel coupling formulation.

    Source: §1 · p. 3

    The analysis assumes that the frequency lies outside the resonance sets; deriving a formulation that removes this assumption is left for future work.

    Source: §6 · p. 15

Abstract

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This paper derives an equivalent formulation of the Costabel FEM-BEM coupling for the time-harmonic Maxwell transmission problem with a Lipschitz interface, and constructs a block-diagonal preconditioner for the resulting discrete system. The reformulated coupling is posed on a product of trace spaces and is obtained by introducing a trace lifting operator and the Steklov--Poincar{\' e} operator. At the algebraic level, it corresponds to eliminating the interior FEM unknowns, resulting in a discrete system that involves the Schur complement of the interior FEM block. Assuming that the frequency lies outside the resonance sets, we prove that the reformulated coupling is equivalent to the Costabel coupling at both the continuous and discrete levels. Using the operator preconditioning framework, we build a preconditioner from the Galerkin matrix induced by the Maxwell single layer boundary integral operator. The spectral condition number of the preconditioned matrix is shown to be uniformly bounded. Numerical experiments illustrate the practical effectiveness of the proposed approach.

1. Introduction

The introduction motivates a Costabel FEM-BEM reformulation and preconditioner for time-harmonic Maxwell scattering with Lipschitz interfaces. Under a resonance exclusion, the reformulated coupling is equivalent to the Costabel coupling, and the preconditioned matrix has a uniformly bounded spectral condition number.

1. Introduction

  • Earlier Maxwell FEM-BEM formulations cited here relied on smooth-interface assumptions, whereas the Costabel coupling has rigorous analysis for Lipschitz interfaces.The cited Costabel results include well-posedness, Galerkin well-posedness, and a quasi-optimal a priori error estimate.Source: §1 · p. 2
  • Direct solvers are impractical for the large sparse-volume and dense-boundary systems, while existing preconditioning analyses face limits on non-smooth surfaces and electromagnetic problems.Arguments relying on positive definiteness or semi-definiteness of FEM matrices in Laplace and Yukawa settings are unavailable for the electromagnetic problem considered here.Source: §1 · p. 2
  • The reformulation uses a trace lifting operator and the Steklov–Poincaré operator to pose the coupling on a product of trace spaces.It is motivated by using the Maxwell single layer boundary integral operator to regularize the Steklov–Poincaré operator.Source: §1 · p. 2
  • The reformulated coupling is equivalent to the Costabel coupling, and the proposed preconditioned matrix has a uniformly bounded spectral condition number when the frequency lies outside the resonance sets.The preconditioner is built from sesquilinear forms involving the Maxwell single layer boundary integral operator and a pairing between discrete spaces.Source: §1 · p. 3

2. Preliminaries

The preliminaries set up the function and trace spaces, operators, and identities used to formulate the Costabel coupling. They also specify the trace framework and boundary integral operators needed for the interior and exterior fields.

2. Preliminaries

  • The paper uses Sobolev spaces of complex vector fields and their natural graph topologies as part of the functional setting.The preliminaries also state that a function's complex conjugate is denoted by an overbar.Source: §2 · p. 3
  • The Dirichlet trace spaces include tangential boundary spaces, and the Dirichlet trace from H(curl, Ωs) is surjective with bounded linear right inverses.The trace pairing is bounded, and the stated trace space is natural for Dirichlet traces of fields in H(curl, Ωs).Source: §2 · pp. 3–4
  • The preliminaries introduce Maxwell single and double layer potentials and their induced boundary integral operators, then recall exterior Calderón identities for radiating solutions.The single and double layer boundary integral operators are denoted Vκ and Kκ.Source: §2 · pp. 4–5

3. Problem reformulation and discretization

This section reformulates the Costabel coupling as an equivalent trace-space variational problem and develops its discrete counterpart under the stated assumptions. It closes the discretization proof by noting bijectivity and continuity and combining estimates.

3.1. Problem reformulation

  • Under Assumption 1(i), the continuous Costabel problem is well-posed, and the discrete Costabel problem is uniquely solvable for sufficiently small h.The continuous result follows from T-coercivity up to a compact perturbation and injectivity; the discrete result assumes sufficiently fine meshes.Source: §3.1 · p. 6, §3.2 · p. 8
  • The trace lifting and induced Steklov–Poincaré operator rewrite the volume term as a boundary pairing, yielding a trace-space formulation equivalent to the Costabel coupling under Assumption 1.The lifting assigns a field to each trace, and the Steklov–Poincaré operator is induced by the variational form.Source: §3.1 · pp. 6–7

3.2. Finite element discretization

  • The discretization assumes a polyhedral domain with quasi-uniform tetrahedral meshes and uses lowest-order Nédélec volume elements with associated lowest-order Raviart–Thomas surface elements.The induced surface meshes inherit shape regularity from the volume meshes.Source: §3.2 · p. 8
  • Under Assumption 1(ii), the discrete lifting is well-posed and uniformly bounded for sufficiently small h, and the discrete Steklov–Poincaré operator is independent of the chosen right inverse.Uniform boundedness is asserted for the lifting operator, while right-inverse independence is stated separately.Source: §3.2 · p. 8
  • For sufficiently small h, the discrete Costabel problem and its reduced trace-space formulation are equivalent under Assumption 1.The equivalence is established for the tetrahedral finite element discretization and its associated surface spaces.Source: §3.2 · pp. 8–9
  • The reduced discrete formulation has an inf-sup constant independent of h for sufficiently small meshes under Assumption 1.Its stability follows from the discrete Costabel formulation and a uniformly bounded discrete right inverse.Source: §3.2 · p. 9
  • The proof notes that two mappings are bijective and invokes continuity of γs.The supplied passage does not identify the mappings fully or state the resulting theorem claim.Source: §3.2 · p. 10
  • The proof concludes by combining (15), an inequality obtained by taking an infimum over finite-element pairs, and a final estimate.The passage identifies these as the steps used to finish the proof but does not provide the estimate values.Source: §3.2 · p. 10

4. Operator preconditioning

This section begins the matrix representation of the discrete reformulated problem and introduces operator preconditioning. It organizes the Galerkin matrix using interior and boundary basis-index groups.

4. Operator preconditioning

  • The discrete problem's matrix representation is the starting point for introducing a preconditioner within the operator preconditioning framework.The framework is cited as [18, Sect. 2].Source: §4 · p. 10
  • The basis functions of Xh are associated with mesh edges, and their indices are partitioned into interior and boundary groups.The index sets are denoted JI and JB, respectively.Source: §4 · p. 10
  • With this index decomposition, the Galerkin matrix associated with a(·, ·) is written in block form.Source: §4 · p. 10

ABI ABB

The construction solves for interior coefficients and substitutes them back into the variational system, using invertibility of AII under Assumption 1(ii).

ABI ABB

  • The basis decomposition separates interior and boundary indices, and the coefficient vector for ζh collects its boundary-basis coefficients.The spaces are decomposed as Xh = Xh,0 ⊕ span{ψi | i ∈ JB}.Source: ABI ABB · p. 11
  • Substituting the solved interior coefficients back into the system and taking Rh = Eh yields the boundary form Sh.The definition of Sh is independent of the choice of Rh.Source: ABI ABB · p. 11
  • Under Assumption 1(ii), well-posedness of Problem (12) makes AII invertible, allowing the interior coefficients to be solved.Source: ABI ABB · p. 11

II AIB

The discrete formulation reduces the system to boundary coefficients, then preconditions it with block-diagonal Galerkin matrices; the analysis gives solvability and a stability result under stated assumptions.

II AIB

  • The Galerkin matrix of Sh is the Schur complement of AII in A, representing elimination of the interior FEM block.Source: II AIB · p. 11
  • The solution of Problem (14) is represented by a coefficient vector for (τh, λh), and the resulting discrete system is uniquely solvable for sufficiently small h under Assumption 1.Source: II AIB · p. 11
  • The Buffa–Christiansen space Zh is built on the barycentric refinement of Γh and has the same dimension as Yh.Source: II AIB · p. 11
  • The preconditioner P for BR is block diagonal and is motivated by regularizing the Steklov–Poincaré operator and operator preconditioning.Source: II AIB · pp. 11–12
  • Theorem 4.1 establishes its stated inequality for sufficiently small h under Assumption 1.Source: II AIB · p. 12
  • The proof relies on uniform stability of the forms, so the same argument applies to preconditioners defined by forms with those stability properties.Source: II AIB · p. 12
  • Preconditioning the reduced system (19) is practical because FEM solvers can apply the Schur complement and solve the sparse recovery system.Source: II AIB · p. 12

5. Numerical results

Across the cuboid and star-based pyramid tests, preconditioning keeps condition numbers stable under mesh refinement and reduces the growth in GMRES iterations. The experiments also show the field across the interface and test the method on a nonconvex polyhedron with reentrant features.

5. Numerical results

  • The experiments use air as the exterior medium at angular frequency ω = 6.28 × 10^8 rad/s, and report condition numbers and GMRES counts across mesh refinements.GMRES uses absolute and relative tolerances of 10^-9.Source: §5 · p. 12

5.1. A cuboid

  • For the cuboid, reformulation modestly lowers condition numbers and GMRES counts, but BR still becomes increasingly ill-conditioned as the mesh is refined.PBR condition numbers remain stable and its GMRES counts vary only mildly across the considered mesh levels.Source: §5.1 · p. 13

5. Numerical results

  • Across both examples, PBR condition numbers remain stable under refinement, while GMRES iterations grow more slowly than for unpreconditioned systems.The comparisons are reported in Tables 1 and 2.Source: Table 1 · p. 13, §5.1 · p. 13, Table 2 · p. 14

5.1. A cuboid

  • The cuboid field reconstruction shows continuity of the tangential x2-component across the interface, consistent with the electric-trace condition.The field component is plotted over a selected region in the plane x2 = 0.5.Source: §5.1 · p. 13, Figure 2 · p. 14

5.2. A star-based pyramid

  • The pyramid test uses a nonconvex polyhedron with sharp reentrant corners and edges, providing a geometrically challenging case.Its base is a 24-pointed star formed from alternating vertices on circles of radii 1 m and 0.3 m.Source: §5.2 · p. 14
  • For the star-based pyramid, BR improves on BC at each mesh level, yet both its condition number and GMRES count grow rapidly under refinement.PBR condition numbers appear stable, and its GMRES counts increase much more slowly and remain far smaller than those of the unpreconditioned system.Source: §5.2 · p. 14, Table 2 · p. 14

6. Conclusions

The paper establishes an equivalent reformulation of the Costabel coupling and proves a uniform bound on the preconditioned matrix's spectral condition number, assuming the frequency lies outside the resonance sets. Removing that frequency assumption and extending the technique to other FEM-BEM couplings are identified as future directions.

6. Conclusions

  • Using a trace lifting operator and the Steklov–Poincaré operator, the reformulated Costabel coupling is equivalent at continuous and discrete levels when the frequency lies outside the resonance sets.The paper constructs a block-diagonal preconditioner and proves a uniformly bounded spectral condition number for the resulting preconditioned matrix.Source: §6 · p. 14
  • Future work includes a regularized formulation to remove the non-resonance assumption and extending the preconditioning technique to other FEM-BEM couplings.Source: §6 · p. 15
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