Volume 10, Issue 2
High Order Hierarchical Divergence-Free Constrained Transport $H(div)$ Finite Element Method for Magnetic Induction Equation

Numer. Math. Theor. Meth. Appl., 10 (2017), pp. 243-254.

Published online: 2017-10

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• Abstract

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.

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@Article{NMTMA-10-243, author = {}, 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 = {

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} }
TY - JOUR T1 - High Order Hierarchical Divergence-Free Constrained Transport $H(div)$ Finite Element Method for Magnetic Induction Equation 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 -

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.

Wei Cai, Jun Hu & Shangyou Zhang. (2020). High Order Hierarchical Divergence-Free Constrained Transport $H(div)$ Finite Element Method for Magnetic Induction Equation. Numerical Mathematics: Theory, Methods and Applications. 10 (2). 243-254. doi:10.4208/nmtma.2017.s03
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