An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincar\'e inequalities, and consistency

TL;DR

This paper introduces a new discrete de Rham complex on polyhedral meshes that is fully discrete, exact, and suitable for nonconforming schemes, with proven stability and error estimates for magnetostatics applications.

Contribution

It develops an arbitrary-order discrete de Rham complex on polyhedral meshes with proven exactness, Poincaré inequalities, and consistency, enabling stable nonconforming PDE schemes.

Findings

Proved exactness and stability of the DDR complex.

Established optimal error estimates for the magnetostatics scheme.

Demonstrated the applicability of the complex to practical PDE problems.

Abstract

In this paper we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into 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, and satisfies suitable exactness properties depending on the topology of the domain. In conjunction with bespoke discrete counterparts of $L^2$-products, it can be used to design schemes for partial differential equations that benefit from the exactness of the sequence but, unlike classical (e.g., Raviart--Thomas--N\'ed\'elec) finite elements, are nonconforming. We prove a complete panel of results for the analysis of such schemes: exactness properties, uniform Poincar\'e 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.