An arbitrary-order discrete de Rham complex on polyhedral meshes. Part II: Consistency

TL;DR

This paper establishes comprehensive consistency results for a discrete de Rham complex on polyhedral meshes, including convergence analysis and numerical validation for magnetostatics, advancing the mathematical foundation of discrete vector calculus.

Contribution

It provides the first complete proof of consistency for the DDR complex, including primal and adjoint operators, and demonstrates its effectiveness through convergence analysis and numerical experiments.

Findings

Proved primal and adjoint consistency of DDR operators.

Established convergence of DDR approximation for magnetostatics.

Validated theoretical results with numerical experiments on polyhedral meshes.

Abstract

In this paper we prove a complete panel of consistency results for the discrete de Rham (DDR) complex introduced in the companion paper [D. A. Di Pietro and J. Droniou, An arbitrary-order discrete de Rham complex on polyhedral meshes. Part I: Exactness and Poincar\'e inequalities, 2021, submitted], including primal and adjoint consistency for the discrete vector calculus operators, and consistency of the corresponding potentials. The theoretical results are showcased by performing a full convergence analysis for a DDR approximation of a magnetostatics model. Numerical results on three-dimensional polyhedral meshes complete the exposition.