A divergence-free finite element method for a type of 3D Maxwell equations

We seek a divergence-free finite element solution for the magnetic field governed by the static Maxwell equations. As usual, the solution is represented as a curl of a vector potential. Typically, this vector potential is uniquely defined in a divergence-free space. The novelty of our method is that we use some simple but non-divergence-free finite element spaces. In this way, the finite element vector potential does not approximate the divergence-free vector, but its curl is divergence-free and is exactly the same solution obtained by the divergence-free finite element potential. Computationally, the finite element solution for the magnetic field is obtained directly as a certain weighted L2-orthogonal projection within the divergence-free finite element subspace. Optimal order convergence is shown for the method. Numerical tests are provided.