@Article{NMTMA-10-243,
author = {Wei Cai, Jun Hu and Shangyou Zhang},
title = {High Order Hierarchical Divergence-Free Constrained Transport $H(div)$ Finite Element Method for Magnetic Induction Equation},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2017},
volume = {10},
number = {2},
pages = {243--254},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.2017.s03},
url = {http://global-sci.org/intro/article_detail/nmtma/12345.html}
}