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

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