Algebraic dual polynomials for the equivalence of curl-curl problems

Abstract

In this paper we will consider two curl-curl equation in two dimensions. One curl-curl problem for a scalar quantity $F$ and one problem for a vector field $\mathbf{E}$. For Dirichlet boundary conditions $\mathbf{n}\times \mathbf{E}=$ ${\hat{E}}_{⊣}$ on $\mathbf{E}$ and Neumann boundary conditions $\bf{n} \times \mbox{curl}$ $F={\hat{E}}_{⊣}$, we expect the solutions to satisfy $\bf{E}=\mbox{curl}$ $F$. When we use algebraic dual polynomial representations, these identities continue to hold at the discrete level. Equivalence will be proved and illustrated with a computational example.

12 Pages, 4 figures