Operator preconditioning of the Costabel FEM-BEM coupling for the time-harmonic Maxwell transmission problem
Abstract
This paper derives an equivalent formulation of the Costabel FEM-BEM coupling for the time-harmonic Maxwell transmission problem with a Lipschitz interface, and constructs a block-diagonal preconditioner for the resulting discrete system. The reformulated coupling is posed on a product of trace spaces and is obtained by introducing a trace lifting operator and the Steklov--Poincar{\' e} operator. At the algebraic level, it corresponds to eliminating the interior FEM unknowns, resulting in a discrete system that involves the Schur complement of the interior FEM block. Assuming that the frequency lies outside the resonance sets, we prove that the reformulated coupling is equivalent to the Costabel coupling at both the continuous and discrete levels. Using the operator preconditioning framework, we build a preconditioner from the Galerkin matrix induced by the Maxwell single layer boundary integral operator. The spectral condition number of the preconditioned matrix is shown to be uniformly bounded. Numerical experiments illustrate the practical effectiveness of the proposed approach.
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BibTeX
@misc{nguyen2026operator,
title = {{Operator preconditioning of the Costabel FEM-BEM coupling for the time-harmonic Maxwell transmission problem}},
author = {Quang Huy Nguyen and Van Chien Le and Kristof Cools},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09765},
eprint = {2610.09765},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09765},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Operator preconditioning of the Costabel FEM-BEM coupling for the time-harmonic Maxwell transmission problem
AU - Quang Huy Nguyen
AU - Van Chien Le
AU - Kristof Cools
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.09765
UR - https://arxiv.org/abs/2610.09765
AB - This paper derives an equivalent formulation of the Costabel FEM-BEM coupling for the time-harmonic Maxwell transmission problem with a Lipschitz interface, and constructs a block-diagonal preconditioner for the resulting discrete system. The reformulated coupling is posed on a product of trace spaces and is obtained by introducing a trace lifting operator and the Steklov--Poincar{\' e} operator. At the algebraic level, it corresponds to eliminating the interior FEM unknowns, resulting in a discrete system that involves the Schur complement of the interior FEM block. Assuming that the frequency lies outside the resonance sets, we prove that the reformulated coupling is equivalent to the Costabel coupling at both the continuous and discrete levels. Using the operator preconditioning framework, we build a preconditioner from the Galerkin matrix induced by the Maxwell single layer boundary integral operator. The spectral condition number of the preconditioned matrix is shown to be uniformly bounded. Numerical experiments illustrate the practical effectiveness of the proposed approach.
ER -