We are interested in exterior boundary value problems (BVP) for the Euclidean Hodge-Helmholtz operator \(-\Delta_{HH}:=\textbf{curl}\;\textbf{curl} -\eta \nabla \text{div} -\kappa^2\), which is closely related to Maxwell equations in frequency domain. In order to solve these exterior BVPs, we consider the corresponding first-kind boundary integral equations, which were derived and analyzed in [2]. It is worth noticing that the boundary integral operators (BIOs) for Hodge-Helmholtz induce bounded and coercive sesquilinear forms in their natural energy trace spaces, and one can establish the unique solvability of the related first-kind boundary integral equations. However, the situation changes when the wavenumber \(\kappa\) is zero, i.e., for the Hodge-Laplacian. Then, the related BIOs feature kernels whose dimensions are linked to fundamental topological invariants of the domain \(\Omega\). Moreover, Galerkin discretization does not affect the dimensions of these kernels [3].

In this talk, we pursue the Galerkin discretization of the variational formulations in [2] and we provide numerical experiments for these boundary integral equations using Bempp [1]. We validate our implementation using a new Calderón residual technique. Then, we compare the eigenvalues related to the equivalent saddle point formulation for \(\kappa\) = 0 with those found in [3] and also present the spectra for small wave numbers \(\kappa\) and discuss its numerical consequences.

References

1. T. Betcke and M. W. Scroggs. Bempp-cl: A fast Python based just-in-time compiling boundary element library Journal of Open Source Software, 6(59), 2879, 2021.

2. X. Claeys and R. Hiptmair. First-kind Boundary Integral Equations for the Hodge-Helmholtz Operator SIAM Journal on Mathematical Analysis, 51(1):197-227, 2019.

3. X. Claeys and R. Hiptmair. First-kind Galerkin boundary element methods for the Hodge-Laplacian in three dimensions Mathematical Methods in the Applied Sciences, 43(8):4974-4994, 2020.

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