Given a sequence of finite element spaces which form a de Rham sequence, we will construct dual representations of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence. The dual representations also need to satisfy the de Rham sequence on the domain boundary. The matrix which converts primal representations to dual representations – the Hodge matrix – is the mass or Gram matrix. It will be shown that a bilinear form of a primal and a dual representation is equal to the vector inner product of the expansion coefficients (degrees of freedom) of both representations. This leads to very sparse system matrices, even for high order methods. The differential operators for dual representations will be defined. Vector operations, grad, curl and div, for primal and dual representations are both topological and do not depend on the metric, i.e. the size and shape of the mesh or the order of the numerical method. Derivatives are evaluated by applying sparse incidence and inclusion matrices to the expansion coefficients of the representations. As illustration of the use of dual representations, the method will be applied to (i) a mixed formulation for the Poisson problem in 3D, (ii) it will be shown that this approach allows one to preserve the equivalence between Dirichlet and Neumann problems in the finite dimensional setting and, (iii) the method will be applied to the approximation of grad–div eigenvalue problem on affine and non-affine meshes.

给定形成de Rham序列的有限元空间序列，我们将使用连接这些空间的关联微分算子构造这些空间的对偶表示，以使它们也形成de Rham序列。对偶表示还需要满足域边界上的de Rham序列。将原始表示转换为对偶表示的矩阵（霍奇矩阵）是质量矩阵或革兰氏矩阵。将显示原始和对偶表示的双线性形式等于两个表示的展开系数（自由度）的向量内积。即使对于高阶方法，这也会导致非常稀疏的系统矩阵。将定义对偶表示的微分运算符。向量运算，包括grad，curl和div，对于原始和对偶表示，它们都是拓扑结构，并且不依赖于度量标准，即网格的大小和形状或数值方法的顺序。通过将稀疏发生率和包含矩阵应用于表示的展开系数来评估导数。作为对偶表示的使用的说明，该方法将应用于（i）3D泊松问题的混合公式，（ii）将显示出这种方法可以保留Dirichlet和Neumann问题之间的等价关系。 （iii）该方法将应用于仿射和非仿射网格上的grad-div特征值问题的逼近。通过将稀疏发生率和包含矩阵应用于表示的展开系数来评估导数。作为对偶表示的使用的说明，该方法将应用于（i）3D泊松问题的混合公式，（ii）将显示出这种方法可以保留Dirichlet和Neumann问题之间的等价关系。 （iii）该方法将应用于仿射和非仿射网格上的grad-div特征值问题的逼近。通过将稀疏发生率和包含矩阵应用于表示的展开系数来评估导数。作为对偶表示的使用的说明，该方法将应用于（i）3D泊松问题的混合公式，（ii）将显示出这种方法可以保留Dirichlet和Neumann问题之间的等价关系。 （iii）该方法将应用于仿射和非仿射网格上的grad-div特征值问题的逼近。