Construction and application of algebraic dual polynomial representations for finite element methods on quadrilateral and hexahedral meshes

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.