Introduction to the Discrete De Rham complex

Jérôme Droniou
Monash university (Melbourne, Australia)

Date(s) : 21/01/2023   iCal
16 h 00 min - 17 h 00 min

Hilbert complexes are chains of spaces linked by operators, with properties that are crucial to establish the well-posedness of certain systems of partial differential equations.
Designing stable numerical schemes for such systems, without resorting to non-physical stabilisation processes, requires reproducing the complex properties at the discrete level.
Finite-element complexes have been extensively developed since the late 2000’s, in particular by Arnold, Falk, Winther and collaborators. These are however limited to certain types of meshes (mostly, tetrahedral and hexahedral meshes), which limits options for, e.g., local mesh refinement.

In this talk we will introduce the Discrete De Rham complex, a discrete version of one of the most popular complexes of differential operators (involving the gradient, curl and divergence), that can be applied on meshes made of generic polytopes.
We will use the Stokes problem in curl-curl formulation to motivate the need for (continuous and discrete) complexes, then give a presentation
of the lowest-order version of the complex. We will then briefly explain how this lowest-order version is naturally extended to an arbitrary-order version,  and present the associated properties (Poincaré inequalities, primal and adjoint consistency, commutation properties, etc.) that enable the analysis of schemes based on this complex. For the Stokes problem we will see that using this complex leads to a pressure-robust scheme.

FRUMAM, St Charles (2ème étage)


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