Algebraic dual polynomials for the equivalence of curl-curl problems
This work provides a theoretical foundation for discrete equivalence in curl-curl problems, which is relevant for computational electromagnetics and related fields.
The paper proves that for two-dimensional curl-curl problems with Dirichlet and Neumann boundary conditions, the continuous identity E = curl F holds at the discrete level when using algebraic dual polynomial representations. Equivalence is demonstrated with a computational example.
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 $\bf{E}$. For Dirichlet boundary conditions $\bf{n} \times \bf{E} =$ $ \hat{E}_{\dashv}$ on $\bf{E}$ and Neumann boundary conditions $\bf{n} \times \mbox{curl}$ $F=\hat{E}_{\dashv}$, 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.