Algebraic dual polynomials for the equivalence of curl-curl problems

TL;DR

This paper establishes the algebraic dual polynomial framework for two-dimensional curl-curl problems, proving the equivalence of scalar and vector formulations at the discrete level with computational validation.

Contribution

It introduces algebraic dual polynomial representations for curl-curl problems and proves their equivalence at the discrete level, supported by computational examples.

Findings

Discrete identities hold for algebraic dual polynomial representations.

Equivalence between scalar and vector curl-curl problems is validated.

Computational example confirms theoretical results.

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 $\bf{E}$. 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.