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High Order Hierarchical Divergence-Free Constrained Transport H(div) Finite Element Method for Magnetic Induction Equation

Part of: Magnetohydrodynamics and electrohydrodynamics Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  09 May 2017

Wei Cai ,
Jun Hu  and
Shangyou Zhang
Wei Cai*
Affiliation:
Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
Jun Hu*
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, China
Shangyou Zhang*
Affiliation:
Department of Mathematics, University of Delaware, Newark, DE 19716, USA
*
*Corresponding author. Email addresses:wcai@uncc.edu (W. Cai), hujun@math.pku.edu.cn (J. Hu), szhang@udel.edu (S. Y. Zhang)
*Corresponding author. Email addresses:wcai@uncc.edu (W. Cai), hujun@math.pku.edu.cn (J. Hu), szhang@udel.edu (S. Y. Zhang)
*Corresponding author. Email addresses:wcai@uncc.edu (W. Cai), hujun@math.pku.edu.cn (J. Hu), szhang@udel.edu (S. Y. Zhang)
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Abstract

In this paper, we propose to use the interior functions of an hierarchical basis for high order BDMp elements to enforce the divergence-free condition of a magnetic field B approximated by the H(div)BDMp 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 BDMp 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.

Keywords

H(div) finite elementshierarchical basisMHDdivergence-free

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76W05: Magnetohydrodynamics and electrohydrodynamics
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 2 , May 2017 , pp. 243 - 254
Copyright
Copyright © Global-Science Press 2017 

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