Given a polynomial basis $\Psi_i$ which spans the polynomial vector space $\mathcal{P}$, this paper addresses the construction and use of the algebraic dual space $\mathcal{P}'$ and its canonical basis. Differentiation of dual variables will be discussed. The method will be applied to a Dirichlet and Neumann problem presented in \cite{CarstensenDemkowiczGopalakrishnan} and it is shown that the finite dimensional approximations satisfy $\phi^h = \mbox{div}\, \mathbf{q}^h$ on any grid. The dual method is also applied to a constrained minimization problem, which leads to a mixed finite element formulation. The discretization of the constraint and the Lagrange multiplier will be independent of the grid size, grid shape and the polynomial degree of the basis functions. 查看全文>>