Abstract
In electromagnetism, a conductor that is not connected to the ground is an equipotential whose value is implicitly determined by the constraint of the problem. It leads to a nonlocal constraints on the flux along the conductor interface, so-called floating potential problems. Unlike previous numerical study that tackle the floating potential problems with the help of advanced and complex numerical methods, we show how an appropriate use of Steklov-Poincaré operators enables to obtain the solution to these partial differential equations with a non local constraint as a linear (and well-designed) combination of \(N+1\) Dirichlet problems, N being the number of conductors not connected to a ground potential. In the case of thin highly conductive inclusions, we perform an asymptotic analysis to approach the electroquasistatic potential at any order of accuracy. In particular, we show t hat the so-called floating potential approaches the electroquasistatic potential with a first order accuracy. This enables us to characterize the configurations for which floating potential approximation has to be used to accurately solve the electroquasistatic problem.
This work was partially funded by the ITMO Cancer in the frame of the Plan Cancer 2014–2019 (project NUMEP PC201615).
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Notes
- 1.
Note that if the inclusion \(\mathcal {O}_k\) is isolated, then \(g_k\) is nothing but 0.
- 2.
To simplify notation, we consider the non dimension conductivity map \(\sigma _{\varepsilon }\), which is the conductivity map divided by the characteristic conductivity of the domain, which might the average of the conductivity on the low conductive domain.
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Collin, A., Corridore, S., Poignard, C. (2021). Floating Potential Boundary Condition in Smooth Domains in an Electroporation Context. In: Suzuki, T., Poignard, C., Chaplain, M., Quaranta, V. (eds) Methods of Mathematical Oncology. MMDS 2020. Springer Proceedings in Mathematics & Statistics, vol 370. Springer, Singapore. https://doi.org/10.1007/978-981-16-4866-3_6
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DOI: https://doi.org/10.1007/978-981-16-4866-3_6
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