Fierro Piccardo, María Ignacia; (2024) Preconditioning of boundary integral equations for Maxwell problems through on-surface radiation conditions. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

The Electric Field Integral Equation (EFIE) is a well-established tool to solve scattering problems. This equation is, however, ill-conditioned under certain circumstances, so developing preconditioning techniques is required to fix this problem. The most common preconditioner for the EFIE is the so-called Calderón preconditioner, that despite its effectiveness (on closed surfaces), can be deemed inefficient due to its implementation on dual meshes, which turns into a six-fold increase in the number of degrees of freedom of the original mesh. Here we study the implementation and the properties of an approximation to an On Surface Radiation Condition operator, first developed in [1] as an approximation of the Magnetic-to-Electric (MtE) map (analogue to the Dirichlet-to-Neumann map in the scalar case) on the primal mesh. Novel results from this study are that the implementation of the MtE on closed surfaces of turns out to be an efficient and reliable preconditioner for the EFIE. Also, on open surfaces, like screens the preconditioner does not need to be modified to keep its preconditioning properties, and it can also be extended to the case of multi-screens, as a diagonal preconditioner. The preconditioning properties of the MtE in the last case is sensitive to dense mesh discretisations, but it will be appropriate to achieve a sensible resolution, at the time it remains robust with respect to increases in the incident wave frequency. We also demonstrate, in a mathematical fashion, that the nature of the preconditioned EFIE using the MtE is a compact approximate to the identity, which should provide good conditioning properties to this formulation. Also, this procedure allowed us to spot the elements that make the approximate MtE a good preconditioner, which might help us to deduce new preconditioning techniques for the EFIE or simplifications to the current ones. We finalise presenting the numerical experiments that helped us to support our theory, showing promising results in all cases.

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