SECOND-ORDER DIVERGENCE-FREE FEM FOR MAGNETOHYDRODYNAMICS USING LAGRANGE POLYNOMIALS

Abstract: AbstractMaxwell’s laws state that the divergence of a magnetic field always remains zero. In thecontinuous setting this is true for the magnetohydrodynamics equations. However, in thediscrete setting, the solution does not always remain divergence-free. The goal of this project isto find two-dimensional finite element meshes for which the magnetic field solution to themagnetohydrodynamics equations remained divergence-free. The report includes the derivationfor a condition of a divergence-free mesh, examples of meshes for which this condition holdstrue, and lastly numerical results to confirm that the solutions using these meshes aredivergence-free.