TY - JOUR
T1 - High Order Hierarchical Divergence-Free Constrained Transport $H(div)$ Finite Element Method for Magnetic Induction Equation
AU - Wei Cai, Jun Hu & Shangyou Zhang
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 243
EP - 254
PY - 2017
DA - 2017/10
SN - 10
DO - http://doi.org/10.4208/nmtma.2017.s03
UR - https://global-sci.org/intro/article_detail/nmtma/12345.html
KW - 
AB - <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>