An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes: Exactness, Poincaré Inequalities, and Consistency

• Published in: Foundations of computational mathematics Vol. 23; no. 1; pp. 85 - 164
• Main Authors: Di Pietro, Daniele A., Droniou, Jérôme
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2023; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Applications of Mathematics; Computer Science; Consistency; Differential equations; Economics; Inequalities; Linear and Multilinear Algebras; Magnetostatics; Math Applications in Computer Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Operators; Partial differential equations; Polynomials; Topology; Discrete de Rham complex; Compatible discretizations; Polyhedral methods; Arbitrary order; 78A30; 65N99; 65N30; Compatible discretisations
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-021-09542-8

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–Nédélec) finite elements, are nonconforming. We prove a complete panel of results for the analysis of such schemes: 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.