论文标题
基于离散DE RHAM序列的多面体网格上的磁静态方法的任意阶
An arbitrary-order method for magnetostatics on polyhedral meshes based on a discrete de Rham sequence
论文作者
论文摘要
在这项工作中，我们开发了一种离散化方法，用于将支持任意顺序和多面部网格的磁静态问题的混合制定。该方法基于一个全局离散的DE RHAM（DDR）序列，该序列是通过在[di pietro，droniou，rapetti，在多边形上任意程度和polyhedra，arxiv：1911.03616]中的任意程度的完全离散的多项式de rham序列来修补构建的局部空间，该元素是通过实现单个元素的元素来实现的。本文的第一个主要贡献是该全局DDR序列的精确性关系证明，以利用相应的局部序列的精确性和网格的拓扑组件的确切性，对无法封闭任何空隙的域有效。第二个主要贡献是该方法的配方和适当性分析，其中包括统一的庞加莱不平等现象的证明，用于离散差异和卷曲操作员。自然能规范中的收敛速率在标准和多面部网格上进行数值评估。当使用$ k \ ge 0 $的DDR序列时，错误将$ h^{k+1} $收敛，而$ h $表示网格size。
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 [Di Pietro, Droniou, Rapetti, Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra, arXiv:1911.03616] 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 meshsize.
