Prof Jerome Droniou

T: +61 3 99054489
E: jerome.droniou@monash.edu

Research Overview

My main interest lies in the design and analysis of numerical methods for partial differential equations of elliptic and parabolic types. My specific areas of interest include:

Polytopal meshes: design of numerical methods that are applicable on meshes made of generic polygons and polyhedra.

High order methods: schemes that possibly have a high order of accuracy even on coarse meshes.

Discrete complexes: numerical techniques that preserve calculus relations of differential complexes, including exactness/cohomology properties.

Convergence analysis: performing the convergence analysis of the schemes, both to capture their (possibly high) accuracy on smooth solutions for simple models, but also using compactness techniques to prove convergence in case of irregular data or solutions, and/or coupled and non-linear models.

Selected Publications

[1] Antonio Pietro, Daniele Di ; Droniou, Jerome. "The Hybrid High-Order Method for Polytopal Meshes : Design, Analysis, and Applications". Cham Switzerland : Springer, 2020. 525 p. (MS&A – Modeling, Simulation and Applications). https://doi.org/10.1007/978-3-030-37203-3 URL: https: //hal.archives-ouvertes.fr/hal-02151813

[2] Droniou, Jerome ; Eymard, Robert ; Gallouet, Thierry ; Guichard, Cindy ; Herbin, Raphaèle. "The Gradient Discretisation Method". Cham Switzerland : Springer, 2018. 497 p. (Mathématiques et Applications). https://doi.org/10.1007/978-3-319-79042-8 URL: https://hal.archives-ouvertes.fr/hal-01382358

[3] Di Pietro, Daniele A.; Droniou, Jérôme. "An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency". In: Found. Comput. Math. 80p, 2021. doi: 10.1007/s10208-021-09542-8. URL: https://arxiv.org/abs/2101.04940.

[4] Cheng, Hanz Martin ; Droniou, Jérôme ; Le, Kim Ngan. "Convergence analysis of a family of ELLAM schemes for a fully coupled model of miscible displacement in porous media". In: Numerische Mathematik. 2019 ; Vol. 141, No. 2. pp. 353–397. https://doi.org/10.1007/s00211-018-1002-2 URL: https://arxiv.org/abs/1710. 01897

[5] Droniou, Jérôme ; Hennicker, Julian ; Masson, Roland. "Numerical analysis of a two-phase flow discrete fracture matrix model". In: Numerische Mathematik. 2019 ; Vol. 141, No. 1. pp. 21–62. https://doi.org/10.1007/s00211-018-0994-y URL: https://arxiv.org/abs/1612.07373