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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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

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