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A HDG Method for Elliptic Problems with Integral Boundary Condition: Theory and Applications


In this paper, we address the study of elliptic boundary value problems in presence of a boundary condition of integral type (IBC) where the potential is an unknown constant and the flux (the integral of the flux density) over a portion of the boundary is given by a value or a coupling condition. We first motivate our work with realistic examples from nano-electronics, high field magnets and ophthalmology. We then define a general framework stemming from the Hybridizable Discontinuous Galerkin method that accounts naturally for the IBC and we provide a complete analysis at continuous and discrete levels. The implementation in the Feel++framework is then detailed and the convergence and scalability properties are verified. Finally, numerical experiments performed on the real-life motivating applications are used to illustrate our methodology.

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Data Availability

Data and meshes for the thermoelectric application are not open: the exact material specifications and geometries are secrets due to the high international competition between USA, Europe, China, Korea and Japan. However close enough models and the software configuration are openly available on Feel++GitHub repository, see The data of the other two applications are openly available in Feel++repository (


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Christophe Prud’homme, Romain Hild and Lorenzo Sala wish to thank Vincent Chabannes from Cemosis for many fruitful discussions. The authors wish to acknowledge the Labex IRMIA from University of Strasbourg for supporting the Eye2Brain project. The PhD thesis of Romain Hild has been also supported by the Labex IRMIA.


Giovanna Guidoboni, Christophe Prud’homme, Lorenzo Sala and Marcela Szopos acknowledge the funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 731063. Giovanna Guidoboni has been partially supported by NSF-DMS 1853222/2021192. She also would like to disclose that she has received remuneration for serving as a consultant for Foresite healthcare LLC.

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Bertoluzza, S., Guidoboni, G., Hild, R. et al. A HDG Method for Elliptic Problems with Integral Boundary Condition: Theory and Applications. J Sci Comput 95, 6 (2023).

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