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

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.

References in corpus (10)

• A multi-block infrastructure for three-dimensional time-dependent numerical relativity
• Arbitrary-order pressure-robust DDR and VEM methods for the Stokes problem on polyhedral meshes
• Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra
• Riemann Normal Coordinates, Smooth Lattices and Numerical Relativity
• Solving Partial Differential Equations Numerically on Manifolds with Arbitrary Spatial Topologies
• An arbitrary order scheme on generic meshes for miscible displacements in porous media
• A numerical study of the Regge Calculus and Smooth Lattice methods on a Kasner cosmology
• Simple Numerical Solutions to the Einstein Constraints on Various Three-Manifolds
• Discrete differential forms for cosmological space-times
• Evolutions of Gowdy, Brill and Teukolsky initial data on a smooth lattice