An arbitrary-order discrete de Rham complex on polyhedral meshes. Part I: Exactness and Poincaré inequalities

Daniele Antonio Di Pietro (1) , Jérôme Droniou (2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 School of Mathematics [Australia]

In this series of papers we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into the ranges of vector calculus operators and complements linked to the spaces in the Koszul complex. The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts. We prove a complete panel of results required for the analysis of discretisation schemes for partial differential equations based on this complex: exactness properties, uniform Poincaré inequalities, as well as primal and adjoint consistency. We also show how this DDR complex enables the design of a numerical scheme for a magnetostatics problem, and use the aforementioned results to prove stability and optimal error estimates for this scheme.

Mots clés: Discrete de Rham complex; Compatible discretisations; Polyhedral methods

Domaines: Analyse numérique [math.NA]

Daniele Antonio Di Pietro, Jérôme Droniou. An arbitrary-order discrete de Rham complex on polyhedral meshes. Part I: Exactness and Poincaré inequalities. 2021. ⟨hal-03103526⟩