Divergence-Free $\boldsymbol{\mathcal{H}}(\mathbf{div})$ -Conforming Hierarchical Bases for Magnetohydrodynamics (MHD)
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Divergence-Free H(div)-Conforming Hierarchical Bases for Magnetohydrodynamics (MHD) Wei Cai · Jian Wu · Jianguo Xin
Received: 5 February 2013 / Accepted: 22 February 2013 / Published online: 27 March 2013 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2013
Abstract In order to solve the magnetohydrodynamics (MHD) equations with a H(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 H(div)-conforming basis. Four such hierarchical bases for the H(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 · H(div)-conforming elements · Divergence-free condition Mathematics Subject Classification (2010) 65N30 · 65F35 · 65F15
1 Introduction The magnetohydrodynamics (MHD) equations describe the dynamics of a charged system under the interaction with a magnetic field and the conservation of the mass, momentum, and energy for the plasma system. Such a dynamics is considered constrained as the magnetic field of the system is evolved with the constraint of zero divergence, namely, ∇ · B = 0. Numerical modeling of plasmas has shown that the observance of the zero divergence of the magnetic field plays an important role in W. Cai () · J. Wu · J. Xin Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA e-mail: [email protected]
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reproducing the correct physics in the plasma fluid [1]. Various numerical techniques have been devised to ensure the computed magnetic field to maintain divergencefree [2]. In the original work of [1] a projection approach was used to correct the magnetic field to have a zero divergence. A more natural way to satisfy this constraint is through a class of the so-called constrained transport (CT) numerical methods based on the ideas in [3]. As noted in [4], a piecewise H(div) vector field on a finite element triangulation of a spatial domain can be a global H(div) field if and only if the normal components on the interface of adjacent elements are continuous. Thus, in most of the CT algorithms for the MHD, the surface averaged magnetic flux over the surface of a 3-D element is used to represent the magnetic field while the volume averaged conserved quantities (mass, momentum, and energy) are used. In the two seminal papers [5, 6], Nédélec proposed to use quantities (moments of normal and tangential components of vector fields) on edges and faces to define the finite dimensional space in H(div) and H
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