We introduce new boundary integral operators which are the exact inverses of the weakly singular and hypersingular operators for −∆ on flat disks. Moreover, we provide explicit closed forms for them and prove the continuity and ellipticity of their corresponding bilinear forms in the natural Sobolev trace spaces. This permit us to derive new Calderón-type identities that can provide the foundation for optimal operator preconditioning in Galerkin boundary element methods.
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 [Formula: see text]. 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 Theory 92 (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 [Formula: see text] duality pairing for scalar functions, instead of the vectorial one. Therefore, this novel approach not only offers [Formula: see text]-independent and [Formula: see text]-robust condition numbers, but it is also easier to implement and accommodates non-uniform meshes without additional computational effort.
We consider first-kind weakly singular and hypersingular boundary integral operators for the Laplacian on screens in R 3 and their Galerkin discretization by means of low-order piecewise polynomial boundary elements. For the resulting linear systems of equations we propose novel Calderón-type preconditioners based on (i) new boundary integral operators, which provide the exact inverses of the weakly singular and hypersingular operators on flat disks, and (ii) stable duality pairings relying on dual meshes. On screens obtained as images of the unit disk under bi-Lipschitz transformations, this approach achieves condition numbers uniformly bounded in the meshwidth even on locally refined meshes. Comprehensive numerical tests also confirm its excellent pre-asymptotic performance.
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