M/F PhD Thesis within the ERC NEMESIS Project

Job Description

Thesis Subject

Discrete de Rham methods (DDR) were introduced in [1,2] to provide discrete counterparts of the de Rham complex on general polytopal meshes. Like most polytopal methods, they require the use of stabilization, whose choice can be delicate and may even lead to computational issues. Recently, using the framework of exterior calculus [3], conforming and explicit liftings have been designed for DDR spaces and operators [4]. The goal of this thesis is to explore the possibility of constructing, from these liftings, polytopal schemes without stabilization, to compare them with existing technologies [5], and to assess whether this stabilization-free method performs better on eigenvalue problems than stabilized methods [6].

[1] D. A. Di Pietro, J. Droniou, and F. Rapetti. Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. Math. Models Methods Appl. Sci., 2020, 30(9):1809-1855. DOI: 10.1142/S0218202520500372
[2] D. A. Di Pietro and J. Droniou. An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency. Found. Comput. Math., 2023, 23:85–164. DOI: 10.1007/s10208-021-09542-8
[3] F. Bonaldi, D. A. Di Pietro, J. Droniou, and K. Hu. An exterior calculus framework for polytopal methods.
J. Eur. Math. Soc., 2025. Published online. DOI: 10.4171/JEMS/1602
[4] Conforming lifting and adjoint consistency for the Discrete de Rham complex of differential forms. D. A. Di Pietro, J. Droniou, and S. Pitassi, 28p, 2025. url: https://arxiv.org/abs/2509.21449.
[5] Lowest order stabilization free virtual element method for the 2D Poisson equation. Berrone S., Borio A., and Marcon F.
Comput. Math. Appl., 177:78–99, 2025.
[6] Approximation of PDE eigenvalue problems involving parameter dependent matrices. Boffi D., Gardini F., and Gastaldi L..
Calcolo, 57(4):Paper No. 41, 21, 2020.

Main Activities:
- Theoretical study of a discrete de Rham complex
- Design and analysis of polytopal numerical schemes
- Implementation in the C++ library HArDCore

Candidates are expected to have a strong background in numerical analysis as well as knowledge of classical partial differential equation models from continuum mechanics. Proficiency in a programming language (preferably C++) will be an additional asset.

Your Work Environment

The recruited PhD student will develop their thesis project within the team of the ERC project NEMESIS (NEw generation MEthods for numerical SImulationS) (erc-nemesis.eu), led at IMAG by two scientific coordinators: Jérôme Droniou (CNRS) and Daniele Di Pietro (University of Montpellier).
Located on the Triolet Campus of the University of Montpellier, IMAG is one of the gateways to mathematics in the Occitanie region. It comprises 170 members and is organized into four research teams: Modélisation, Analyse et Calcul Scientifique (MACS), Didactique et Epistémologie des Mathématiques (DEMA), Équipe de Probabilité et Statistique (EPS), and Géométrie, Topologie et Algèbre (GTA).

Compensation and benefits

Compensation

2300 € gross monthly

Annual leave and RTT

44 jours

Remote Working practice and compensation

Pratique et indemnisation du TT

Transport

Prise en charge à 75% du coût et forfait mobilité durable jusqu’à 300€

About the offer

About the CNRS

The CNRS is a major player in fundamental research on a global scale. The CNRS is the only French organization active in all scientific fields. Its unique position as a multi-specialist allows it to bring together different disciplines to address the most important challenges of the contemporary world, in connection with the actors of change.

CNRS

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