A polytopal discrete de Rham complex on manifolds, with application to the Maxwell equations

TL;DR

This work tackles stable, high-order discretizations of the de Rham complex on manifolds by introducing a polytopal, locally defined discrete de Rham (DDR) complex that preserves the topological structure of $H^k(\Omega)$ while accommodating generic curved mesh elements. It constructs local polynomial spaces via affine-compatible charts and a metric-aware Hodge star, proving an isomorphism of discrete and continuous cohomologies and establishing primal consistency in an intrinsic, manifold setting. The authors then apply the framework to a 2+1 Maxwell system on curved 2D manifolds (sphere and torus), designing a semi-discrete scheme that preserves the discrete constraint and energy, and they validate the method with both smooth and non-smooth solutions. Numerical results show high-order convergence on curved geometries, with energy and constraint preservation demonstrated to machine precision, highlighting the approach’s potential for geometry-aware electromagnetism and other PDEs on manifolds.

Abstract

We design in this work a discrete de Rham complex on manifolds. This complex, written in the framework of exterior calculus, has the same cohomology as the continuous de Rham complex, is of arbitrary order of accuracy and, in principle, can be designed on meshes made of generic elements (that is, elements whose boundary is the union of an arbitrary number of curved edges/faces). Notions of local (full and trimmed) polynomial spaces are developed, with compatibility requirements between polynomials on mesh entities of various dimensions. We give explicit constructions of such polynomials in 2D, for some meshes made of curved triangles or quadrangles (such meshes are easy to design in many cases, starting from a few charts describing the manifold). The discrete de Rham complex is then used to set up a scheme for the Maxwell equations on a 2D manifold without boundary, and we show that a natural discrete version of the constraint linking the electric field and the electric charge density is satisfied. Numerical examples are provided on the sphere and the torus, based on bespoke analytical solutions and meshes on each of these manifolds.

Figures (18)

• Figure 1: Mesh used to discretise one chart with its orientation.
• Figure 2: Embedding of the $1$-skeleton of the mesh into $\mathbb{R}^3$.
• Figure 3: Finer meshing of the disk.
• Figure 4: Embedding of the finer mesh into $\mathbb{R}^3$.
• Figure 6: Error on $E$ on the sphere.

Theorems & Definitions (57)

• Definition 1: Equivalence of meshes
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6: Decomposition of polynomial spaces
• proof
• Definition 7: Trimmed polynomial spaces
• Remark 8
• Definition 10: Local discrete exterior derivative and discrete potential