In this work we develop a discretisation method for the mixed formulation of the magnetostatic problem supporting arbitrary orders and polyhedral meshes. The method is based on a global discrete de Rham (DDR) sequence, obtained by patching the local spaces constructed in [22] by enforcing the single-valuedness of the components attached to the boundary of each element. The first main contribution of this paper is a proof of exactness relations for this global DDR sequence, obtained leveraging the exactness of the corresponding local sequence and a topological assembly of the mesh valid for domains that do not enclose any void. The second main contribution is the formulation and well-posedness analysis of the method, which includes the proof of uniform Poincaré inequalities for the discrete divergence and curl operators. The convergence rate in the natural energy norm is numerically evaluated on standard and polyhedral meshes. When the DDR sequence of degree $k\ge 0$ is used, the error converges as $h^{k+1}$, with h denoting the mesh size.

在这项工作中，我们为支持任意阶次和多面体网格的静磁问题的混合公式开发了离散化方法。该方法基于全局离散de Rham（DDR）序列，该序列是通过强制附加到每个元素边界的分量的单值修补在[22]中构造的局部空间而获得的。本文的第一个主要贡献是证明了此全局DDR序列的精确度关系，它是利用相应局部序列的精确度以及对不包含任何空隙的域有效的网格拓扑组合而获得的。第二个主要贡献是该方法的公式化和适度分析，其中包括离散散度和卷曲算子的一致庞加莱不等式的证明。在标准和多面体网格上对自然能范数的收敛速度进行了数值评估。当DDR顺序度$ķ\ge 0$ 使用时，误差收敛为 $H^{ķ+\mathrm{1个}}$，其中h表示网格尺寸。