Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T09:50:28.677Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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:[email protected] (W. Cai), [email protected] (J. Hu), [email protected] (S. Y. Zhang)
*Corresponding author. Email addresses:[email protected] (W. Cai), [email protected] (J. Hu), [email protected] (S. Y. Zhang)
*Corresponding author. Email addresses:[email protected] (W. Cai), [email protected] (J. Hu), [email protected] (S. Y. Zhang)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ainsworth, M. and Coyle, J., Hierarchic finite element bases on unstructured tetrahedral meshes, Int. J. Numer. Methods Eng., 58 (2003), pp. 2103–2130.Google Scholar
[2] Cai, W., Wu, J. and Xin, J. G., Divergence-free H(div)-conforming hierarchical bases for magnetohydrodynamics (MHD), Commun. Math. Stat., 1 (2013), pp. 19–35.Google Scholar
[3] Brackbill, J. U. and Barnes, D. C., The effect of nonzero product of magnetic gradient and B on the numerical solution of the magnetohydrodynamic equations, J. Comput. Phys., 35 (1980), pp. 426–430.Google Scholar
[4] Brezzi, F., Douglas, J., Duran, R. and Fortin M, M., Mixed finite elements for second order elliptic problems in three variables, Numerische Mathematik, 51(2) (1987), pp. 237–50.CrossRefGoogle Scholar
[5] Evans, C. R. and Hawley, J. F., Simulation of magnetohydrodynamic flows: a constrained transport method, Astrophys. J., 332 (1988), pp. 659–677.CrossRefGoogle Scholar
[6] Tóth, G., The ∇ · B = 0 constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161 (2000), pp. 605–652.Google Scholar
[7] Hu, J., Finite element approximations of symmetric tensors on simplicial grids in Rn: the higher order case, J. Comput. Math., 33 (2015), pp. 283–296.CrossRefGoogle Scholar
[8] Hu, J., A new family of efficient conforming mixed finite elements on both rectangular and cuboid meshes for linear elasticity in the symmetric formulation, SIAM J. Numer. Anal., 53 (2015), pp. 1438–1463.Google Scholar
[9] Hu, J. and Zhang, S., A family of symmetric mixed finite elements for linear elasticity on tetredral grids, Sci. China Math., 58 (2015), pp. 297–307.Google Scholar