We design in this work a discrete de Rham complex on manifolds. This complex, written in the framework of exterior calculus, is applicable on meshes on the manifold with generic elements, and has the same cohomology as the continuous de Rham complex. Notions of local (full and trimmed) polynomial spaces are developed, with compatibility requirements between polynomials on mesh entities of various dimensions. Explicit examples of polynomials spaces are presented. 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 a bespoke analytical solution and mesh design on each manifold.

翻译：本文设计了一个流形上的离散de Rham复形。该复形在外微分框架下构建，适用于流形上具有一般单元的网格，且与连续de Rham复形具有相同的上同调。我们发展了局部（完全与截断）多项式空间的概念，并规定了各维网格实体上多项式之间的相容性条件，给出了多项式空间的具体实例。随后将离散de Rham复形应用于无界二维流形上Maxwell方程的求解，证明了电场与电荷密度之间约束关系的自然离散版本得以满足。基于各流形上特制的解析解和网格设计，提供了在球面和环面上的数值算例。