Multigrid computation of vector potentials
Journal of Computational and Applied Mathematics
Covolume Solutions of Three-Dimensional Div-Curl Equations
SIAM Journal on Numerical Analysis
Discrete Vector Potentials for Nonsimply Connected Three-Dimensional Domains
SIAM Journal on Numerical Analysis
Locally divergence-free discontinuous Galerkin methods for the Maxwell equations
Journal of Computational Physics
A Family of $Q_{k+1,k}\timesQ_{k,k+1}$ Divergence-Free Finite Elements on Rectangular Grids
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
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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 L^2-orthogonal projection within the divergence-free finite element subspace. Optimal order convergence is shown for the method. Numerical tests are provided.