DivergenceFree \(\boldsymbol{\mathcal{H}}(\mathbf{div})\)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 divergencefree 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 Divergencefree conditionMathematics Subject Classification (2010)
65N30 65F35 65F151 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 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 socalled constrained transport (CT) numerical methods based on the ideas in [3]. As noted in [4], a piecewise \(\boldsymbol{\mathcal{H}}(\mathbf{div})\) vector field on a finite element triangulation of a spatial domain can be a global \(\boldsymbol{\mathcal{H}}(\mathbf{div})\) 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 3D 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 \(\boldsymbol{\mathcal{H} }(\mathbf{div})\) and \(\boldsymbol{\mathcal{H}}(\mathbf{curl})\), respectively. The specific construction of the basis functions in both spaces, specifically in \(\boldsymbol{\mathcal{H}}(\mathbf{div})\), can be done in various ways such as the hierarchical type of basis proposed in [7] for both \(\boldsymbol{\mathcal{H}}(\mathbf{div})\) and \(\boldsymbol{\mathcal{H}}(\mathbf{curl})\). Unfortunately, the proposed hierarchical basis for \(\boldsymbol{\mathcal{H}}(\mathbf{div})\) in [7] turns out to be erroneous as can be easily checked; for the quadratic polynomial approximation of the proposed edgebased basis functions happen to be linearly dependent.
In this paper, we will first present a new hierarchical basis functions in \(\boldsymbol{\mathcal{H}}(\mathbf{div})\) in the tetrahedral case, for completeness, together with a review of the hierarchical basis of \(\boldsymbol{\mathcal{H}}(\mathbf{div})\) for rectangles in 2D and for a cube in 3D. An important common feature of the hierarchical basis functions is the fact that for order p≥4 in the case of simplexes, the basis functions will include interior bubble basis functions which have zero normal components on the whole boundary of the element. Therefore, a simple way to enforce the property of being divergencefree can be easily accomplished by adding one single (p+1)th order (any qth order, q≥max(m,p+1),m=4 for tetrahedral element, m=3 for triangular element, and m=2 for quadrilateral or hexahedral element) interior bubble function on each element to a pth order \(\boldsymbol{\mathcal{H}}(\mathbf{div})\) basis. This extra bubble basis will be able to satisfy the local divergencefree condition. Due to the fact of normal continuity of the pth order basis across the element interface and zero normal component of the addedin qth order interior bubble functions, the augmented function space satisfies the divergencefree condition globally. The \(\boldsymbol{\mathcal{H}}(\mathbf{div})\) basis on general meshes other than the reference elements mentioned above are usually constructed by a Piola transform [8] and more recent studies of the \(\boldsymbol{\mathcal{H}}(\mathbf{div})\) basis functions on general quadrilateral and hexahedral elements can be found in [8, 9, 10], however, they will not be discussed further in this paper.
The rest of the paper is organized as follows. The constructions of the \(\boldsymbol{\mathcal{H}}(\mathbf{div})\) bases are given in Sects. 2–5 for rectangular and triangular elements in 2D, and cubic and tetrahedral elements in 3D. The divergencefree condition is discussed in Sect. 6. Numerical results on the matrix conditioning are given in Sect. 7. Concluding remarks are given in Sect. 8.
2 Basis Functions for the Quadrilateral Element
In [11], Zaglmayr gave a hierarchical basis for quadrilateral \(\boldsymbol{\mathcal{H}}(\mathbf{div})\)conforming element. In this section we summarize the result in [11].
2.1 EdgeBased Functions
These functions are further grouped into two categories: the lowestorder and higherorder functions.
LowestOrder Functions
HigherOrder Functions
2.2 Interior Functions
The interior functions are classified into three categories.
Type 1 (Curl Field)
Type 2
Type 3
Decomposition of the space Q _{ p+1,p }×Q _{ p,p+1} for the \(\boldsymbol{\mathcal{H}}(\mathbf{div})\)conforming quadrilateral element
Decomposition  Dimension 

Edgebased functions  4(p+1) 
Interior functions  2p(p+1) 
Total  2(p+2)(p+1)=Dim(Q _{ p+1,p }×Q _{ p,p+1}) 
3 Basis Functions for the Hexahedral Element
3.1 FaceBased Functions
In this subsection we record the results in [11]. We have also fixed one error in [11]. These functions are associated with the six faces whose formulas are classified into two groups.
LowestOrder Raviart–Thomas Functions
HigherOrder Functions (DivergenceFree)
3.2 Interior Functions
The interior functions are further classified into three categories. The triplet (ξ _{1},η _{2},ζ _{3}):=(2ξ−1,2η−1,2ζ−1) is used in the formulas. The function ℓ _{ n }(•) is the classical unnormalized Legendre polynomial of degree n. While we here record the results in [11], we have implemented the correction of a number of mistakes in [11] as well.
Type 1 (DivergenceFree)
Type 2
Type 3

All the interior basis functions are linearly independent, which can be verified easily.

The normal traces of these interior functions vanish on the boundary \(\partial\mathcal{H}\) of the reference hexahedral element. This is due to the fact that on a certain face f either one of the standard unit vectors is perpendicular to the normal vector of the face n _{ f } or to the fact that the integrated Legendre polynomials are evaluated at 1 or −1, which leads to 0.
Decomposition of the space \((\mathbb{Q}_{n}(K))^{3}\) for the \(\boldsymbol{\mathcal{H}}(\mathbf{div})\)conforming hexahedral element
Decomposition  Dimension 

Facebased face functions (lowest order RT)  6 
Facebased face functions (higher order)  6p(p+2) 
Interior functions  3p(p+1)^{2} 
Total  3(p+2)(p+1)^{2}=dimQ _{ p+1,p,p }×Q _{ p,p+1,p }×Q _{ p,p,p+1} 
4 Basis Functions for the Triangular Element
The result on the basis construction has been reported in [12]. For the completeness of the current study, we record the basis functions in this section.
4.1 Edge Functions
4.2 Interior Functions
EdgeBased Interior Functions
Interior Bubble Functions
Decomposition of the space \(( \mathbb{P}_{p}(K))^{2}\) for the \(\boldsymbol{\mathcal{H}}(\mathbf{div})\)conforming triangular element
Decomposition  Dimension 

Edge functions  3(p+1) 
Edgebased interior functions  3(p−1) 
Interior bubble functions  (p−2)(p−1) 
Total  (p+1)(p+2)=dim(P _{ p }(K))^{2} 
5 Basis Functions for the Tetrahedral Element
Our constructions are motivated by the work on the construction of \(\boldsymbol{\mathcal{H}}(\mathbf{div})\)conforming hierarchical bases for tetrahedral elements [7]. We construct shape functions for the \(\boldsymbol{\mathcal{H}}(\mathbf{div})\)conforming tetrahedral element on the canonical reference 3simplex. The shape functions are grouped into several categories based upon their geometrical entities on the reference 3simplex [7]. The basis functions in each category are constructed so that they are orthonormal on the reference element.
The standard bases in \(\mathbb{R}^{n}\) are noted as \(\vec{e}_{i}\), i=1,…,n, and n={2,3}.
5.1 Face Functions
The face functions are further grouped into two categories: edgebased face functions and face bubble functions.
EdgeBased Face Functions

First kind highorder independent edgebased face functions

Second kind highorder independent edgebased face functions
An alternative approach using the idea of recursion from [7] can also be used to construct independent edgebased face functions as follows.
Face Bubble Functions
5.2 Interior Functions
EdgeBased Interior Functions
FaceBased Interior Functions
Interior Bubble Functions
Decomposition of the space \(( \mathbb{P}_{p}(K))^{3}\) for the \(\boldsymbol{\mathcal{H}}(\mathbf{div})\)conforming tetrahedral element
Decomposition  Dimension 

Edgebased face functions  12p 
Face bubble functions  2(p−2)(p−1) 
Edgebased interior functions  6(p−1) 
Facebased interior functions  4(p−2)(p−1) 
Interior bubble functions  (p−3)(p−2)(p−1)/2 
Total  (p+1)(p+2)(p+3)/2=dim(P _{ p }(K))^{3} 
6 The DivergenceFree Condition
7 Conditioning of Matrices
The purpose of this section is twofold. Firstly, we check numerically that the newly constructed basis functions for \(\boldsymbol{\mathcal{H}}(\mathbf{div} )\)conforming triangular and tetrahedral elements are linearly independent, which is manifested by the fact that, for each particular approximation order up to degree four, the condition number of the corresponding mass matrix is finite. Secondly, we want to show that for the approximation up to order three both the mass and the stiffness matrices are reasonably wellconditioned.
Condition numbers of the mass matrix M and stiffness matrix S from the basis for the \(\boldsymbol{\mathcal{H}}(\mathbf{div})\)conforming triangular element
Order p  Mass  Stiffness 

1  2.016e1  1.040e1 
2  8.804e1  5.959e1 
3  9.847e2  4.197e2 
4  1.286e4  8.843e3 
From the table we can see that the condition number is bounded for each order of approximation. Moreover, up to the fourth order, the mass and stiffness matrices are both well conditioned.
Condition numbers of the mass matrix M and stiffness matrix S from the bases with two different kinds of edgebased face function for the \(\boldsymbol{\mathcal{H}}(\mathbf{div})\)conforming tetrahedral element
Order p  Mass  Stiffness  Ratio  

First kind  Second kind  First kind  Second kind  Mass  Stiff.  
1  3.084e1  3.084e1  1.989e1  1.989e1  1.000e0  1.000e0 
2  6.987e3  7.733e4  3.395e3  5.917e4  0.090e0  0.057e0 
3  3.412e6  2.289e6  1.094e6  1.191e6  1.491e0  0.919e0 
4  5.972e9  2.717e7  2.883e9  2.372e7  2.198e2  1.215e2 
Again, from this table we see that the condition number is finite for each order of approximation. Further up to the third order, both the mass and the stiffness matrices are well conditioned. For order p=2 the conditioning is better with the first kind edgebased face basis, while for p=4, the conditioning is better with the second kind edgebased face basis. For the third order p=3, the performance with both kinds of edgebased face basis is about the same.
8 Concluding Remarks
In this paper we focus our attention on hierarchical \(\boldsymbol{\mathcal{H}}(\mathbf{div})\) basis functions for solving the magnetohydrodynamics (MHD) equations numerically so that the divergencefree condition on the magnetic field is rigorously guaranteed. The idea is to use an interior bubble function from the proposed highorder hierarchical basis as the additional freedom to impose the divergencefree Gauge condition for the magnetic field. We have summarized four bases for the \(\boldsymbol{\mathcal{H} }(\mathbf{div})\)conforming elements, viz. the quadrilateral and triangular elements for 2D and the hexahedral and tetrahedral elements for 3D. The linear independence of the basis functions for the two simplicial elements has numerically been checked. Good matrix (mass and stiffness) conditioning has also been shown up to the fourth order for 2D and up to the third order for 3D. Further work will include the implementation of the proposed divergencefree basis to solve the magnetohydrodynamics (MHD) equations in 2D and 3D.
Notes
Acknowledgements
This research is supported in part by a DOE grant DEFG0205ER25678 and a NSF grant DMS1005441.
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