$\boldsymbol{\mathcal{H}}(\mathbf{div})$-conforming elements,Divergence-free condition" /> $\boldsymbol{\mathcal{H}}(\mathbf{div})$-conforming elements" /> $\boldsymbol{\mathcal{H}}(\mathbf{div})$-conforming elements,Divergence-free condition" />

Divergence-Free $\boldsymbol{\mathcal{H}}(\mathbf{div})$-Conforming Hierarchical Bases for Magnetohydrodynamics (MHD)

Wei Cai , Jian Wu , Jianguo Xin

Communications in Mathematics and Statistics ›› 2013, Vol. 1 ›› Issue (1) : 19 -35.

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Communications in Mathematics and Statistics ›› 2013, Vol. 1 ›› Issue (1) : 19 -35. DOI: 10.1007/s40304-013-0003-9
Original Article

Divergence-Free $\boldsymbol{\mathcal{H}}(\mathbf{div})$-Conforming Hierarchical Bases for Magnetohydrodynamics (MHD)

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Abstract

In order to solve the magnetohydrodynamics (MHD) equations with a $\boldsymbol{\mathcal{H}}(\mathbf{div})$-conforming element, a novel approach is proposed to ensure the exact divergence-free condition on the magnetic field. The idea is to add on each element an extra interior bubble function from a higher order hierarchical $\boldsymbol{\mathcal{H}}(\mathbf{div})$-conforming basis. Four such hierarchical bases for the $\boldsymbol{\mathcal{H}} (\mathbf{div})$-conforming quadrilateral, triangular, hexahedral, and tetrahedral elements are either proposed (in the case of tetrahedral) or reviewed. Numerical results have been presented to show the linear independence of the basis functions for the two simplicial elements. Good matrix conditioning has been confirmed numerically up to the fourth order for the triangular element and up to the third order for the tetrahedral element.

Keywords

Hierarchical bases / $\boldsymbol{\mathcal{H}}(\mathbf{div})$-conforming elements')">$\boldsymbol{\mathcal{H}}(\mathbf{div})$-conforming elements / Divergence-free condition

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Wei Cai, Jian Wu, Jianguo Xin. Divergence-Free $\boldsymbol{\mathcal{H}}(\mathbf{div})$-Conforming Hierarchical Bases for Magnetohydrodynamics (MHD). Communications in Mathematics and Statistics, 2013, 1(1): 19-35 DOI:10.1007/s40304-013-0003-9

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