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

Daniele A. Di Pietro, Jérôme Droniou|arXiv (Cornell University)|Jan 13, 2021

Advanced Numerical Methods in Computational Mathematics54 references29 citations

TL;DR

This paper presents an arbitrary-order discrete de Rham complex on polyhedral meshes that ensures exactness, uniform Poincaré inequalities, and consistency for compatible finite element-like schemes. By decomposing polynomial spaces using vector calculus operators and Koszul-type complements, the method constructs nonconforming, fully discrete spaces and operators that preserve the topological and geometric structure of the continuous de Rham complex, enabling optimal error estimates for magnetostatics 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.

Motivation & Objective

• To develop a compatible, nonconforming finite element method on general polyhedral meshes that preserves the exactness of the de Rham complex.
• To ensure uniform Poincaré inequalities for discrete norms across mesh sizes.
• To establish primal and adjoint consistency of discrete vector calculus operators with respect to interpolation.
• To provide a framework for constructing stable and optimally convergent schemes for PDEs such as magnetostatics.
• To generalize existing Raviart-Thomas and Nédélec elements to arbitrary order and polyhedral elements using a unified algebraic and geometric decomposition.

Proposed method

• Decomposes polynomial spaces into ranges of vector calculus operators (grad, curl, div) and orthogonal complements via the Koszul complex.
• Defines discrete spaces on mesh entities (cells, faces, edges) with degrees of freedom tied to polynomial moments.
• Constructs discrete vector calculus operators (gradient, curl, divergence) using local reconstruction and lifting techniques.
• Introduces discrete L2-products via potential reconstructions and component-wise norms to ensure stability and consistency.
• Employs local lifting operators (e.g., curl lifting) to prove boundedness and commutation properties.
• Uses scaling arguments and inverse inequalities to derive uniform bounds and Poincaré-type estimates.

Experimental results

Research questions

• RQ1 Can a fully discrete de Rham complex be constructed on general polyhedral meshes that preserves exactness at the discrete level?
• RQ2 Do the discrete operators satisfy uniform Poincaré inequalities independent of mesh size?
• RQ3 Are the discrete operators consistent with their continuous counterparts in both primal and adjoint forms?
• RQ4 Can the discrete complex be used to design a stable and optimally convergent scheme for a magnetostatics problem?
• RQ5 How can polynomial decompositions via vector calculus operators and Koszul complements be leveraged to build arbitrary-order nonconforming methods?

Key findings

• The proposed discrete de Rham complex achieves exactness: Im(𝑖Ω) = Ker(grad), Im(grad) = Ker(curl), Im(curl) = Ker(div), and Im(div) = L2(Ω), under topological assumptions.
• Uniform discrete Poincaré inequalities are proven: ‖u‖L2(T) ≲ h_T ‖grad u‖L2(T) for u ∈ (grad H1(T))⊥, and analogous bounds for curl and div.
• Primal consistency holds: the discrete gradient, curl, and divergence operators commute with interpolation operators up to O(h^k) terms.
• Adjoint consistency is established: the discrete operators satisfy integration-by-parts identities with optimal consistency error O(h^k).
• The discrete L2-products are equivalent to the continuous L2-inner product, ensuring stability and boundedness of the discrete norms.
• For a magnetostatics problem, the scheme achieves optimal convergence: ‖B - B_h‖L2(Ω) ≲ h^k, with k the polynomial degree, under appropriate regularity.