In this paper, we propose to use the interior functions of an hierarchical basis for high order $BDM_p$ elements to enforce the divergence-free condition of a magnetic field $B$ approximated by the $H(div)$ $BDM_p$ basis. The resulting constrained finite element method can be used to solve magnetic induction equation in MHD equations. The proposed procedure is based on the fact that the scalar ($p-1$)-th order polynomial space on each element can be decomposed as an orthogonal sum of the subspace defined by the divergence of the interior functions of the $p$-th order $BDM_p$ basis and the constant function. Therefore, the interior functions can be used to remove element-wise all higher order terms except the constant in the divergence error of the finite element solution of the $B$-field. The constant terms from each element can be then easily corrected using a first order $H(div)$ basis globally. Numerical results for a 3-D magnetic induction equation show the effectiveness of the proposed method in enforcing divergence-free condition of the magnetic field.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2017.s03}, url = {http://global-sci.org/intro/article_detail/nmtma/12345.html} }