Divergence-free H(div)-conforming hierarchical bases for magnetohydrodynamics (MHD)

In order to solve the magnetohydrodynamics (MHD) equations with a $\mathbf{\mathcal{H}}(\mathbf{div})$-conforming element, a novel approach is proposed to ensure the exact divergence-free condition on the magnetic field. The idea is to add on each element an extra interior bubble function from higher order hierarchical $\mathbf{\mathcal{H}}(\mathbf{div})$-conforming basis. Four such hierarchical bases for the $\mathbf{\mathcal{H}}% (\mathbf{div})$-conforming quadrilateral, triangular, hexahedral and tetrahedral elements are either proposed (in the case of tetrahedral) or reviewed. Numerical results have been presented to show the linear independence of the basis functions for the two simplicial elements. Good matrix conditioning has been confirmed numerically up to the fourth order for the triangular element and up to the third order for the tetrahedral element.