The electric field integral equation can describe scattering by closed and open surfaces, surfaces containing junctions, and even non-orientable surfaces. The boundary element discretisation of this equation results in linear systems whose condition number grows as the square of the inverse mesh size. This eventually leads to systems that in practice cannot be solved, not even when using powerful iterative solvers such as GMRES and efficient matrix compression algorithms such as the fast multipole algorithm or an H-matrix based low rank representation. As a remedy, Calderón preconditioners are used to significantly reduce the number of iterations required to reach an acceptable solution. This type of preconditioners are available for open and closed surfaces, and recently also for surfaces containing junctions. In this contribution, a Calderón type preconditioner will be constructed for the electric field integral equation applied to non-orientable surfaces such as the Moebius strip. It is based on a redundant representation for the induced current, and a block-diagonal preconditioning strategy. Numerical experiments corroborate the correctness and efficiency of this approach.@en

In this contribution a novel fast-converging integral equation method is introduced that can be used to solve the mixed transmission/scattering problems by composite structures including multiple domains and metallic coatings that can contain junctions. This is achieved by combining the global multi-trace method with the quotient space discretisation of the multi-screen boundary integral equation. The method is illustrated by means of a sufficiently general geometry, the discretisation is discussed, and an effective Calderon multiplicative preconditioner is introduced. Numerical results corroborate the correctness and efficiency of the method.@en

In this contribution, a well-conditioned method for the modelling of scattering by so-called multi-screens or PEC sheets including junctions is introduced. The method starts from the inflated screen approach by Claeys and Hiptmair. We introduce a Calderón preconditioner and a suitable discretisation scheme. The resulting scheme contains many more DoFs than strictly required. We will show how almost all redundancy can be removed without significant loss of effectiveness of the method.@en

We present a new approach for boundary integral equations for the wave equation with zero initial conditions. Unlike previous attempts, our mathematical formulation allows us to prove that the associated boundary integral operators are continuous and satisfy inf-sup conditions in trace spaces of the same regularity, which are closely related to standard energy spaces with the expected regularity in space and time. This feature is crucial from a numerical perspective, as it provides the foundations to derive sharper error estimates and paves the way to devise efficient adaptive space-time boundary element methods, which will be tackled in future work. On the other hand, the proposed approach is compatible with the current time dependent boundary element method's implementations, and we predict that it explains many of the behaviors observed in practice but that were not understood with the existing theory.

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We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain Ω, where Ω is either in Rⁿ or in a Riemannian manifold. For linear systems of equations arising from low-order Galerkin discretizations, we obtain condition numbers that are independent of the mesh size and of the choice of bases for test and trial functions. The basic ingredient is a classical formula by Boggio for the fractional Laplacian, which is extended analytically. In the special case of the weakly and hypersingular operators on a line segment or a screen, our approach gives a unified, independent proof for a series of recent results by Hiptmair, Jerez-Hanckes, Nédélec and Urzúa-Torres. We also study the increasing relevance of the regularity assumptions on the mesh with the order of the operator. Numerical examples validate our theoretical findings and illustrate the performance of the proposed preconditioner on quasi-uniform, graded and adaptively generated meshes.

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A complex screen is an arrangement of panels that may not be even locally orientable because of junction lines. A comprehensive trace space framework for first-kind variational boundary integral equations on complex screens has been established in Claeys and Hiptmair (Integr Equ Oper Theory 77:167–197, 2013. https://doi.org/10.1007/s00020-013-2085-x) for the Helmholtz equation, and in Claeys and Hiptmair (Integr Equ Oper Theory 84:33–68, 2016. https://doi.org/10.1007/s00020-015-2242-5) for Maxwell’s equations in frequency domain. The gist is a quotient space perspective that allows to make sense of jumps of traces as factor spaces of multi-trace spaces modulo single-trace spaces without relying on orientation. This paves the way for formulating first-kind boundary integral equations in weak form posed on energy trace spaces. In this article we extend that idea to the Galerkin boundary element (BE) discretization of first-kind boundary integral equations. Instead of trying to approximate jumps directly, the new quotient space boundary element method employs a Galerkin BE approach in multi-trace boundary element spaces. This spawns discrete boundary integral equations with large null spaces comprised of single-trace functions. Yet, since the right-hand-sides of the linear systems of equations are consistent, Krylov subspace iterative solvers like GMRES are not affected by the presence of a kernel and still converge to a solution. This is strikingly confirmed by numerical tests.

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We construct inverses of the variational electric field boundary integral operator up to compact perturbations on orientable topologically simple screens. We describe them as solution operators of variational problems set in low-regularity standard trace spaces. On flat disks these variational problems do not involve the inversion of any non-local operators. This result lays the foundation for operator preconditioning for the discretized electric field integral equation.@en

Tools Share Abstract We consider the electric field integral equation (EFIE) modeling the scattering of time-harmonic electromagnetic waves at a perfectly conducting screen. When discretizing the EFIE by means of low-order Galerkin boundary methods (BEM), one obtains linear systems that are ill-conditioned on fine meshes and for low wave numbers k . This makes iterative solvers perform poorly and entails the use of preconditioning. In order to construct optimal preconditioners for the EFIE on screens, the authors recently derived compact equivalent inverses of the EFIE operator on simple Lipschitz screens in [R. Hiptmair and C. Urzúa-Torres, Compact equivalent inverse of the electric field integral operator on screens, Integral Equations Operator Theory92 (2020) 9]. This paper elaborates how to use this result to build an optimal operator preconditioner for the EFIE on screens that can be discretized in a stable fashion. Furthermore, the stability of the preconditioner relies only on the stability of the discrete L2 duality pairing for scalar functions, instead of the vectorial one. Therefore, this novel approach not only offers h-independent and k-robust condition numbers, but it is also easier to implement and accommodates non-uniform meshes without additional computational effort. @en