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Quasi-Helmholtz Calder\'on Multiplicative Preconditioning for Higher-Order Global Multi-Trace Integral Equations

Cedric Münger, Alessandro Zuccotti, Van Chien Le, Kristof Cools

arXiv:2609.17252v1cs.CEcs.NAmath.NA

TL;DR

Higher-order global multi-trace Calderón preconditioning is difficult to construct with explicit dual basis functions, and higher-order projector computation also poses implementation challenges. The paper uses quasi-Helmholtz projectors and iterative projector actions to build and solve a preconditioned system; numerical results report mesh- and basis-order-independent iteration counts and low-frequency stability.

  • Problem

    For higher-order bases, constructing explicit dual functions for Calderón preconditioning is challenging, while higher-order projector matrices complicate standard algebraic-multigrid computation.

    Source: §I · p. 1, §V · p. 3

  • Method

    The paper builds a higher-order global multi-trace Calderón preconditioner with quasi-Helmholtz projectors and computes projector actions iteratively for use with GMRES.

    Source: §I · p. 1, §VIII · p. 10

  • Results

    Numerical results report preconditioned iteration counts independent of mesh size and basis order, and stable higher-order multi-trace methods at very low frequencies.

    Source: §VIII · p. 10

  • Takeaways & Limitations

    The approach applies higher-order Calderón preconditioning without dual basis functions and uses standard GMRES without explicitly computing projector pseudo-inverses.

    Source: §VIII · p. 10

    For sphere validation, flat-triangle geometric error is the primary error source, so higher-order basis functions do not yield lower error there.

    Source: A. Validation of solution · p. 7

Abstract

from arXiv · show

The paper presents a higher-order global multi-trace integral equation for time-harmonic electromagnetic scattering by composite objects. The higher-order multi-trace formulation is preconditioned with a Calder\'on multiplicative preconditioner using higher-order quasi-Helmholtz projectors. The higher-order quasi-Helmholtz projectors separate the solenoidal and non-solenoidal components of the basis functions. Separate access to the Helmholtz components circumvents the explicit inversion of an ill-conditioned mixed Gram matrix. This enables the application of Calder\'on multiplicative preconditioning without refining the mesh and resorting to dual basis functions. Furthermore, it enables low-frequency stabilization, as the solenoidal and non-solenoidal components can be rescaled individually. The higher-order quasi-Helmholtz projectors are computed iteratively, enabling iterative solvers to efficiently solve the preconditioned matrix system, yielding accurate solutions in a few dozen iterations. Numerical experiments are conducted for composite dielectric bodies, confirming the effectiveness of the proposed preconditioner for the global multi-trace formulation discretized with higher-order basis functions for dense meshes and at very low frequencies

I. Introduction M

The paper extends Calderón multiplicative preconditioning to higher-order global multi-trace formulations using quasi-Helmholtz projectors. This approach avoids explicit inversion of an ill-conditioned mixed Gram matrix and does not require mesh refinement or dual basis functions.

I. Introduction M

  • Higher-order basis functions make explicit dual-space construction for Calderón preconditioning more challenging than in the zeroth-order case.The passage compares higher-order dual functions to Buffa–Christiansen functions used with Rao–Wilton–Glisson functions.Source: §I · p. 1
  • Helmholtz decomposition separates surface currents into solenoidal and nonsolenoidal components, which higher-order projectors map between basis spaces of different orders.This discrete duality avoids explicitly inverting an ill-conditioned mixed Gram matrix.Source: §I · p. 1
  • Higher-order quasi-Helmholtz projectors enable block-diagonal Calderón preconditioners for global multi-trace discretizations without explicit mixed-Gram inversion or dual basis functions.The proposed projector-action method uses standard tools and can be combined with GMRES.Source: §I · p. 1

II. Global Multi-Trace Formulation

  • The global multi-trace formulation extends PMCHWT to multiple dielectric domains, using electric and magnetic surface currents and incident fields.The exposition restricts itself to two touching domains but states that the formulation extends to any number.Source: §II · p. 2

III. Discrete higher-Order Quasi-Helmholtz Projectors

This section introduces discrete higher-order quasi-Helmholtz projectors built around higher-order basis spaces and Gram matrices. A second projector pair is identified as relevant to low-frequency stability for multiply connected geometries.

III. Discrete higher-Order Quasi-Helmholtz Projectors

  • The order-p quasi-Helmholtz projector is defined using the higher-order basis and its Gram matrix.The GWP basis functions are div-conforming; the Gram matrix entries are L2 inner products of basis functions.Source: §III · p. 2
  • The static contribution Kp|0 is the κ → 0 contribution of the double-layer operator.The section also introduces another projector pair based on the generalized loop matrix.Source: §III · p. 2
  • A second pair of quasi-Helmholtz projectors is used later to develop a low-frequency-stable formulation for multiply connected geometries.The projector properties are introduced alongside this role.Source: §III · p. 2

IV. Quasi-Helmholtz Calderón Multiplicative

The higher-order Calderón multiplicative preconditioner pairs primal GWP spaces with a higher-order dual space and maps their Helmholtz components separately. Its block structure supports extension to an arbitrary number of domains.

IV. Quasi-Helmholtz Calderón Multiplicative

  • The dual space is chosen at order r = p + 2, a choice reported as optimal for the EFIE.The passage contrasts this approach with constructing higher-order dual functions on a barycentrically refined mesh.Source: §IV · p. 3
  • The discrete duality pairing maps order-p solenoidal and nonsolenoidal components separately onto the opposite components of order-r GWP functions.The primal solenoidal subspace must match or exceed the dual nonsolenoidal subspace, and the primal nonsolenoidal subspace must match the dual solenoidal subspace in dimension.Source: §IV · p. 3
  • The mixed Gram matrix Nrp pairs order-r GWP test functions with rotated order-p GWP trial functions.The discretized formulation also uses Maxwell single- and double-layer operators with rotated order-p test functions and order-p trial functions.Source: §IV · p. 3
  • The component mappings are assembled into the preconditioner Tr for the higher-order multi-trace system.The discretized block-preconditioned system is stated to extend readily to an arbitrary number of domains.Source: §IV · p. 3

V. Implementation Details

The implementation addresses the difficulty of applying standard algebraic multigrid to higher-order projectors, whose matrices are no longer sparse. It computes projector actions through saddle-point problems that can be solved with standard iterative methods.

V. Implementation Details

  • Higher-order projector construction involves pseudo-inverting generalized star and loop matrix products, but the inverse Gram matrix makes the relevant matrix nonsparse for standard algebraic multigrid.Specialized higher-order algebraic multigrid methods exist, but are not widely available as black-box solvers in standard numerical tools.Source: §V · p. 3
  • The projectors are recast as equivalent saddle-point problems that can be solved efficiently with an appropriate preconditioner and a standard iterative solver such as GMRES.This avoids resorting to algebraic multigrid.Source: §V · p. 3

A. Iterative Computation of Projectors

  • When the full system is solved iteratively, only each projector’s action on a coefficient vector is needed, rather than an explicit projector matrix.Explicit matrices would require pseudo-inversion; the same saddle-point approach also computes the complementary projector’s action.Source: A. Iterative Computation of Projectors · p. 3
  • A Schur-complement computation verifies that the saddle-point problem’s right-hand side evaluation is equivalent to applying the projector to the coefficient vector.The passage describes this equivalence for the projector action on j.Source: A. Iterative Computation of Projectors · p. 3

B. Solving the Saddle-Point Problem

The saddle-point systems are handled with block-diagonal preconditioning and sparse-matrix techniques. The implementation supports iterative solution while reusing a one-time Cholesky factorization where appropriate.

B. Solving the Saddle-Point Problem

  • Saddle-point systems with this structure also occur in mixed Poisson and Darcy discretizations, where iterative preconditioners can avoid an expensive pseudoinverse.The cited discussion describes inversion in only a few iterations rather than computing the pseudoinverse.Source: B. Solving the Saddle-Point Problem · p. 4
  • A block-diagonal preconditioner proposed in is used to evaluate the saddle-point problem iteratively.The saddle-point matrix is preconditioned with the inverse of a block-diagonal matrix.Source: B. Solving the Saddle-Point Problem · p. 4
  • The matrix Pp is the Gram matrix of discontinuous Lagrange basis functions of order p, with entries defined by their inner products.The accompanying entry expression uses the divergence of the GWP basis functions.Source: B. Solving the Saddle-Point Problem · p. 4
  • For the loop-based projectors, the saddle-point problem can also be computed iteratively, using another preconditioner for its saddle-point matrix.The inverse of a specified matrix is reported to work well for that system.Source: B. Solving the Saddle-Point Problem · p. 4
  • Sparse symmetric matrices in Pdiv and Pcurl are factorized once by Cholesky, then solved using forward and backward substitutions during iterations.The blockwise inverse is computed directly from the factorization.Source: B. Solving the Saddle-Point Problem · p. 4
  • The matrices are defined per domain, so explicit Cholesky may be less economical when domain sizes differ substantially.For one or more much larger domains, another approach to reduce computational cost may be more economical.Source: B. Solving the Saddle-Point Problem · p. 4

VI. Low Frequency Stabilazation

The low-frequency analysis finds that the quasi-Helmholtz CMP higher-order global multi-trace formulation remains stable as frequency tends to zero. The section addresses low-frequency breakdown through rescaling and relates the system blocks to Helmholtz components and operator scaling.

VI. Low Frequency Stabilazation

  • The Helmholtz decomposition separates the basis into local loops, stars, and harmonic functions, represented by loop, star, and discrete harmonic matrices.The loop-star transformation is introduced to express this decomposition.Source: §VI · p. 4
  • The scaling analysis covers single- and double-layer operators and, for plane-wave excitation, the right-hand side.Source: §VI · p. 4
  • Diagonal and off-diagonal blocks have identical low-frequency scaling, and the shown 2 × 2 block scales similarly to PMCHWT.The off-diagonal contributions are related to the low-frequency analysis of MFIE.Source: §VI · p. 5
  • Integral equations including EFIE, MFIE, and PMCHWT are known to suffer low-frequency breakdown, and breakdown occurs as κ approaches 0.Quasi-Helmholtz projectors combined with CMP are described as stable and well-conditioned at very low frequencies.Source: §VI · pp. 4–5
  • The low-frequency breakdown is addressed by rescaling the coefficients and block preconditioner Tr directly in the preconditioned system.The passage contrasts this direct stabilization with separately rescaling the preconditioner and system before combining them.Source: §VI · p. 5
  • The quasi-Helmholtz CMP higher-order global multi-trace formulation remains stable in the low-frequency limit.The section identifies this as the outcome after applying the modified preconditioner and rescaling the system blocks.Source: §VI · p. 5

VII. Numerical Results

The numerical experiments assess the correctness and efficiency of the higher-order multi-trace preconditioning strategy. They test orders p = 0 through 3 and compare against existing formulations and analytic Mie series where possible.

VII. Numerical Results

  • Experiments assess the correctness and efficiency of the preconditioning strategy for higher-order multi-trace integral equations.Source: §VII · p. 5
  • The qH-CMP multi-trace method is evaluated with basis orders p = 0, 1, 2, and 3.Source: §VII · p. 5
  • Comparisons include zeroth-order PMCHWT, multi-trace formulations preconditioned with Buffa-Christiansen dual functions, and analytic Mie series where possible.Source: §VII · p. 5

A. Validation of solution

The unit sphere was split into two unchanged-material domains to validate multi-trace solutions against Mie-series and PMCHWT results. The numerical solutions agree with the references across surface-current, near-field, and radar-cross-section comparisons, while flat-triangle geometry error limits the sphere’s higher-order accuracy gains.

A. Validation of solution

  • The sphere’s one-quarter/three-quarter split leaves both domains’ material properties unchanged, allowing multi-trace results to be compared with the whole-sphere Mie solution and PMCHWT.This simple sphere also establishes a baseline for comparisons on more complex geometries.Source: A. Validation of solution · p. 5, Figure 2 · p. 5
  • The PMCHWT and lowest-order multi-trace surface currents match the analytic Mie-series solution.The multi-trace case uses a two-domain sphere, with the outer-surface mesh held the same across all three comparisons.Source: Figure 3 · p. 6, A. Validation of solution · p. 6
  • All numerical near-field solutions had roughly 1.5% relative error at the sphere center and 0.5% 1 m from its surface versus Mie.The comparison includes PMCHWT and multi-trace orders 0–3.Source: Figure 4 · p. 6, A. Validation of solution · p. 6
  • For far-field bistatic RCS, PMCHWT RMS error was 6.44% and multi-trace ranged from 5.85% to 7.28% relative to Mie; zeroth order was most accurate.The plotted RCS solutions for PMCHWT and multi-trace orders 0–3 match the Mie-series reference.Source: Figure 5 · p. 6, A. Validation of solution · p. 6
  • On the sphere, flat-triangle geometric error is the primary error source, explaining why higher-order basis functions do not yield lower error.The sphere’s curved geometry is approximated by flat triangles.Source: A. Validation of solution · p. 7

B. Convegence

For two touching boxes, higher-order multi-trace solutions generally achieve lower errors on coarser meshes, though sharp geometry limits convergence and higher orders increase system size. At roughly the same number of unknowns, higher orders provide better or comparable accuracy.

B. Convegence

  • For p = 0, 1, and 2, energy-norm errors converge at roughly rate 1; sharp box features constrain higher-order convergence, despite generally lower errors on coarser meshes.The higher-order methods can therefore require fewer degrees of freedom and produce smaller systems.Source: B. Convegence · p. 7

C. Impact of higher-order basis

  • Higher-order solutions are significantly closer to the reference, and can be more accurate on a coarse mesh than lower-order solutions on a finer mesh.Errors decrease with mesh refinement at all orders; the reference is a fine-mesh, order-3 solution.Source: C. Impact of higher-order basis · p. 7, Figure 8 · p. 7, Figure 9 · p. 8
  • At the same mesh size, unknowns rise by approximately 3.5× from order 0 to 1, 7× to order 2, and more than 12× to order 3.The larger higher-order systems are more computationally expensive to assemble and solve.Source: C. Impact of higher-order basis · p. 8, Table I · p. 8
  • At roughly the same number of unknowns, higher-order bases deliver better or equivalent accuracy, corresponding to similar computational effort.The higher-order accuracy gain is visible when relative errors are grouped by basis order.Source: C. Impact of higher-order basis · p. 8, Figure 9 · p. 8

D. Conditioning and iterative solution

The higher-order multi-trace system is solved with GMRES, using iterative projector actions within the preconditioned formulation. The preconditioner sharply reduces iteration counts, which remain relatively low across mesh sizes and basis orders in the reported examples.

D. Conditioning and iterative solution

  • The projector action is computed by solving a saddle-point problem during GMRES, rather than explicitly forming projector matrices via a pseudo-inverse.The action must be computed repeatedly because the preconditioned formulation uses multiple projectors.Source: 1) Higher order projectors · p. 8
  • With the cited preconditioning strategy, projector-action solves take fewer than 10 iterations, independent of unknown count and basis order.Projector setup takes less than 5% of total solution time for third-order problems and less than 3% for zeroth-order problems.Source: 1) Higher order projectors · p. 8, Table III · p. 8
  • At moderate frequency and relatively low contrast, GMRES converges in approximately 60 iterations with the quasi-Helmholtz Calderón multiplicative preconditioner, versus over 5,000 iterations for the regular system with third-order basis functions.For the preconditioned system, the iteration count is independent of mesh size and basis-function order.Source: 2) Higher-order multi-trace formulation · p. 9
  • For boxes with permittivities 6 and 10, iteration counts rise overall versus the lower-contrast example but remain relatively low, with slight variation across mesh sizes and basis orders.Table IV reports the preconditioned-system iteration counts at relative tolerance 10^-8.Source: 2) Higher-order multi-trace formulation · p. 9, Table IV · p. 9

E. Very low frequency

Very-low-frequency tests on multiply connected geometries show agreement between the multi-trace solution and reference solutions, while the conclusion reports mesh- and order-independent iteration counts. Higher-order basis functions reduce error at increased computational cost, and geometric error can obscure their benefits on smooth geometries.

E. Very low frequency

VIII. Conclusion

  • For the preconditioned system, iteration counts vary little across mesh resolutions, indicating that mesh size does not affect solver convergence in the reported results for geometries with global loops.Table V reports a multiply connected geometry with a zeroth-order multi-trace formulation at very low frequency (κ0 = 10−40).Source: E. Very low frequency · p. 9, §VIII · p. 10, Table V · p. 10
  • Higher-order basis functions can significantly reduce error, but require more time and computational resources.The conclusion presents increased computational cost as the trade-off for reduced error.Source: §VIII · p. 10
  • For multiply connected domains, only small modifications to higher-order multi-trace formulations are required for stability at very low frequencies.The tests include a sphere with four central blocks removed and two adjoining cubes with holes through all faces.Source: E. Very low frequency · p. 9, Figure 12 · p. 9, §VIII · p. 10
  • On smooth geometries, geometric error can quickly dominate and obscure higher-order basis benefits; curved triangles or NURBS could improve the approximation.The paper identifies better geometric approximation as a possible improvement.Source: §VIII · p. 10
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