A Finite-Element Framework for a Mimetic Finite-Difference ...

A FINITE-ELEMENT FRAMEWORK FOR A MIMETICFINITE-DIFFERENCE DISCRETIZATION OF MAXWELL’S

EQUATIONS∗

JAMES H. ADLER† , CASEY CAVANAUGH† , XIAOZHE HU† , AND LUDMIL T.

ZIKATANOV ‡

Abstract. Maxwell’s equations are a system of partial differential equations that govern thelaws of electromagnetic induction. We study a mimetic finite-difference (MFD) discretization ofthe equations which preserves important underlying physical properties. We show that, after mass-lumping and appropriate scaling, the MFD discretization is equivalent to a structure-preservingfinite-element (FE) scheme. This allows for a transparent analysis of the MFD method using theFE framework, and provides an avenue for the construction of efficient and robust linear solvers forthe discretized system. In particular, block preconditioners designed for FE formulations can beapplied to the MFD system in a straightforward fashion. We present numerical tests which verifythe accuracy of the MFD scheme and confirm the robustness of the preconditioners.

Key words. Maxwell’s equations, finite-element method, mimetic finite-difference method,structure-preserving block preconditioners

AMS subject classifications. 35Q61, 65M60, 65M06, 65F08, 65Z05

1. Introduction. We consider the numerical solution of Maxwell’s equations ina bounded connected domain Ω ∈ R3:

∂B

∂t+∇×E = 0, in Ω× (0, T ],(1.1)

ε∂E

∂t−∇× µ−1B = −j, in Ω× (0, T ],(1.2)

∇ · εE = 0, in Ω× (0, T ],(1.3)

∇ ·B = 0, in Ω× (0, T ],(1.4)

Here, B(x, t) and E(x, t) are the unknown magnetic and electric fields, ε(x) andµ(x) are the permittivity and permeability of the medium, respectively, and j(x, t)is the current density satisfying ∇ · j = 0. For simplicity, we choose ε = µ = 1, andimpose homogeneous essential (Dirichlet) boundary conditions which model a perfectconductor:

(1.5) B · n∣∣∂Ω

= 0, n×E∣∣∂Ω

= 0.

More general cases can be handled with straightforward modifications. In particular,the analysis presented in this paper still holds with non-constant ε(x) and µ(x), forexample, by using a piecewise constant approximation for the parameters.

The coupled equations (1.1)–(1.2), Faraday’s and Ampere’s laws, model the inter-action of the electric and magnetic fields, while the Gauss laws (1.3)–(1.4), model theflux constraints of the individual fields. One of the major difficulties in numericallysolving the Maxwell system is related to the constraints (1.3) and (1.4) as it is often

∗Submitted to the editors November ??, 2020.†Department of Mathematics, Tufts University, Medford, MA 02155 ([email protected],

[email protected], [email protected]).‡Department of Mathematics, Pennsylvania State University, University Park, PA 16802

([email protected]).

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2 J. H. ADLER, C. CAVANAUGH, X. HU, L.T. ZIKATANOV

necessary, by physical or other considerations, to have analogues of such identities onthe discrete level.

To rectify this, structure-preserving discretizations are used to guarantee thatconservation laws on the continuous level still hold at the discrete level. While thereare a variety of such numerical methods, we focus in this paper on the relationshipbetween mimetic finite-differences (MFD) and a structure-preserving mixed finite-element method (FEM) derived via finite-element exterior calculus (FEEC).

The MFD method is defined by operators designed to “mimic” the continuouslevel operators [8, 19, 22, 31]. This technique is straightforward to derive and, bydrawing similarities to the continuous operators, it is quite easy to see that the discreteoperators do in fact obey the continuous level properties. MFD is simple to implement,can be applied directly to the strong form of the PDE system, and has few meshrequirements (i.e., general polyhedral grids can be used). Like most finite-differencemethods, however, the error estimates and convergence theory require high regularity;well-posedness is difficult to prove; and it is not always clear how to optimally solvethe resulting linear system.

On the other hand, structure-preserving finite-element methods for Maxwell’sequations have been widely studied (eg. [2, 3, 26, 27]). Using FEEC [4, 16], one canshow that de Rham exact sequences are guaranteed on the discrete level. Furthermore,in the FEEC setting, the convergence theory and a priori error estimates are well-known, and showing well-posedness of the discretized weak form follows directly fromBabuska–Brezzi theory [5, 12]. Moreover, the construction of efficient solvers is alsowell-developed [9, 10, 23, 24, 25, 28, 27].

An important question that arises when using such discretizations for PDEs ishow to efficiently solve the resulting linear system. One widely used method for solv-ing Maxwell’s system discretized by FEM is based on block preconditioners for Krylovmethods, constructed using Schur complements. More generally, block precondition-ers for saddle point systems, a class that the full Maxwell system falls into, are widelystudied [9, 10, 23, 24, 25], and robustness and efficiency results are well-established.In general, to ensure that the iterative solver does not destroy the properties of thediscretization, preconditioners must be developed that also preserve the operator prop-erties at each time step. This is essential for ensuring that the resulting numericalsolution obeys the PDE constraints throughout both the spatial and time domains.Such preconditioners have been developed for the full Maxwell system, (1.1)–(1.4),using mixed FEM discretizations [2, 27], as well as for the simplified time-harmonicform of Maxwell’s equations [15, 32]. Another variety of electromagnetic applicationscomes in the form of magnetohydrodynamics, where a similar block-preconditioningapproach can be used for a FEM divergence-free preserving discretization [28].

In this work, we adapt the preconditioners developed in [2] for the Maxwell systemwith impedance boundary conditions. As in [2, 3], we consider a variation of (1.1)–(1.4), where an auxiliary pressure variable, p(x, t), is introduced:

∂B

∂t+∇×E = 0, in Ω× (0, T ],(1.6)

∂E

∂t−∇×B +∇p = −j, in Ω× (0, T ],(1.7)

∂p

∂t+∇ ·E = 0, in Ω× (0, T ].(1.8)

It is straightforward to show that with suitable initial conditions, p(x, 0) = 0 and∇ ·B(x, 0) = 0, (1.1)–(1.4) is equivalent to (1.6)–(1.8). A structure preserving dis-

FE FRAMEWORK OF MFD FOR MAXWELL 3

cretization is essential to allowing this form to be solved in place of the full Maxwellsystem.

Ideally, we would like to have the ease and simplicity of the MFD method withall of the well-posedness and preconditioning theory that supports the FEM. Similarto [11, 29], in this paper, we apply the MFD method to the Maxwell system, andthen analyze it in a FE framework. By using mass-lumping schemes and scaledbasis functions in the FEM, we show that the two methods yield equivalent linearsystems. This equivalence is used to apply the FE theory to the MFD system toshow well-posedness of the mimetic discretization. Additionally, since robust blockpreconditioners have been designed for the Maxwell FE system that preserve theconstraints at all time iterations [2], we demonstrate how to slightly modify thoseresults to obtain robust linear solvers for the MFD system of Maxwell’s equations.

This paper is organized as follows. In Section 2, we introduce the notation anddiscretization technique for MFD and apply it to the Maxwell system. Section 3recalls the FE discretization and presents a mass-lumped alternative. Then, Section4 draws connections between the two methods for Maxwell, shows their equivalence,and well-posedness of the MFD system is proven. Section 5 introduces and analyzesblock preconditioners for the MFD system, proving their robustness. Finally, Section6 presents numerical results to demonstrate the theoretical results, and concludingremarks and future work are discussed in Section 7.

2. The Mimetic Finite-Difference Method. Following [31], we construct aprimal (Delaunay) tetrahedral mesh and a dual (Voronoi) polyhedral grid. Denotethe vertices/nodes of the Delaunay triangulation by xDi

NDi=1, and an edge on the

Delaunay mesh connecting nodes xDi and xDj by eDij , with unit tangent vector tDijpointing from vertex of lower index to vertex of higher index. The Delaunay tetrahedraare given by Dk, k = 1, ..., NV . Each Dk has face set (boundary) ∂Dk. The neighborset of tetrahedron Dk, given by ND

k , is defined as the set of indices of the tetrahedrathat share common planes with Dk, i.e., ND

k := m∣∣ ∂Dk∩∂Dm 6= ∅, m = 1, ..., NV .

The common plane (face) between Dk and Dm is given by ∂Dkm. For tetrahedron Dk

with face ∂Dkm, we define the unit outward normal vector nDkm pointing outward fromDk. Analogous to the above definitions, we have for the dual Voronoi mesh, nodes,xVk

NV

k=1, edges, eVkm, with unit tangent vector tVkm, polyhedra, Vi, i = 1, ..., ND, faceset, ∂Vi, neighbor set, N V

i , common plane ∂Vij , and outward unit normal vector nVij .This dual mesh configuration yields some useful properties that are exploited

when defining the MFD operators. The Voronoi point xVk is the circumcenter ofthe Delaunay tetrahedron Dk. Additionally, the Delaunay nodes define the Voronoipolyhedra. We define Vi as the set of points in the domain that lie closer to Delaunaynode xDi than any other Delaunay node,

Vi := x ∈ Ω∣∣ |x− xDi | ≤ |x− xDj |, j = 1, ..., ND, j 6= i.

Furthermore, we have that each Delaunay edge eDij is orthogonal to the Voronoi face

∂Vij , and each Voronoi edge eVkm is orthogonal to Delaunay face, ∂Dkm. This gives usa one-to-one correspondence between nodes on one mesh to polyhedra on the other,and edges on one mesh to faces on the other. Figure 1 illustrates an example meshin two dimensions to further highlight the notation used. Note that this dual meshconfiguration with these properties requires that the circumcenters of the Delaunaytriangulation lie in the interior of the Delaunay tetrahedra. While this is not a strictrequirement for the MFD method to work, it allows for simplicity in the analysis andimplementation (see [13, 31]).


xD1 xD2

xD5

xD7xD6

xD4 xD3

xD8

xD10

xD9

xV3

xV2

xV1

xV10

xV9

xV8

xV5

xV6

xV4

xV7

xD1

xD4 xD3

eD14 eD13

eD34

xV2

xD2

xV3

xV8

xV5

xV6

xV4

xV7

eV78

eV34

eV67

eV45

eV38 eV56

Fig. 1: Top: Two-dimensional primal Delaunay mesh in black solid lines with corre-sponding dual Voronoi mesh in red dashed lines. Bottom Left: Zoom in of Delaunayelement D2, corresponding to Voronoi node xV2 , with labeled Delaunay edges andnodes. Bottom Right: Zoom in of Voronoi element V2, corresponding to Delaunaynode xD2 , with labeled Voronoi nodes and edges.

2.1. Grid Functions and MFD Operators. Following [19, 31], we definefunctions and operators on both the Delaunay and Voronoi meshes. First, approxi-mations of scalar functions in the domain are represented with scalar grid functionsthat are defined on the nodes of the meshes. Thus, scalar functions defined on theDelaunay nodes are constants on Voronoi polyhedra, and scalar functions on Voronoinodes are constants on Delaunay tetrahedra. The corresponding function spaces are


as follows,

HD := u(x)∣∣u(x) = u(xDi ) = uDi , ∀x ∈ Vi, i = 1, ..., ND,(2.1)

HV := u(x)∣∣u(x) = u(xVk ) = uVk , ∀x ∈ Dk, k = 1, ..., NV .(2.2)

Vector functions are approximated on the Delaunay mesh with vector grid functions,where the function space is denoted by HD. For vector function u(x), we project italong the Delaunay edge, and evaluate at the intersection of the Delaunay edges andVoronoi face. The space of vector grid functions on the Voronoi mesh, HV , is definedanalogously on the Voronoi mesh,

HD := u(x)∣∣u(x) = u · tDij(xDij) = uDij , x

Dij = eDij ∩ ∂Vij,(2.3)

HV := u(x)∣∣u(x) = u · tVkm(xVkm) = uVkm, x

Vkm = eVkm ∩ ∂Dkm.(2.4)

To build intuition, we first introduce the MFD operators component-wise, thenlater give the matrix definitions. In the continuous setting, the gradient maps scalarfunctions to vector functions. Analogously, the discrete gradient on the Delaunaymesh maps a scalar grid function defined on the nodes to a vector grid functiondefined on the edges, or gradDu : HD →HD. On edge eDij ,

(gradDu)Dij =uDj − uDi|eDij |

η(i, j),

where η is an orientation constant,

η(i, j) =

1, j > i

−1, otherwise.

Similarly, the gradient on the Voronoi mesh, gradV u : HV →HV , is given by,

(gradV u)Vkm =uVm − uVk|eVkm|

η(k,m).

To define the discrete divergence, first note that divergence maps vector functions toscalar functions. This differential operator on the Delaunay grid, divDu : HD → HD,corresponding to the outward flux of Vi is defined as

(divDu)Di =1

|Vi|∑j∈NV

i

|∂Vij |uDij(nVij · tDij).

Similarly, on the Voronoi grid, the divergence divV u : HV → HV is

(divV u)Vk =1

|Dk|∑

m∈NDk

|∂Dkm|uVkm(nDkm · tVkm).

The discrete curl operator maps from edges on one mesh (the circulation) to edgeson the other mesh (the axis of rotation) by the geometric relationships between thedual meshes. Therefore, the Delaunay curl operator maps vector grid functions on theDelaunay mesh to a vector grid function on the Voronoi mesh, curlDu : HD → HV

and is given by,


(curlDu)Vkm =(tVkm · nDkm)

|∂Dkm|∑

eDij∈∂Dkm

uDij |eDij | χ(nDkm, tDij),

where the constant χ essentially enforces the right-hand rule,

χ(nDkm, tDij) =

1, tDij positively oriented,

−1, otherwise.

Similarly, we define the Voronoi curl operator curlV u : HV →HD,

(curlV u)Dij =(tDij · nVij)|∂Vij |

∑eVkm∈∂Vij

uVkm |eVkm| χ(nVij , tVkm).

To define the MFD operators in matrix form, we introduce the edge-vertex signedincidence matrix, G ∈ RMD×ND , and the face-edge signed incidence matrix, K ∈RMV ×MD . Both are defined on the Delaunay triangulation whereND, MV andMD de-note the number of Delaunay nodes, Voronoi edges, and Delaunay edges, respectively.Similarly, on the Voronoi mesh, we have the signed incidence matrix GV ∈ RMV ×NV ,where NV denotes the number of nodes on the Voronoi mesh. The nonzero entries ofG, GV , and K are either 1 or −1, and the signs are consistent with the pre-determinedorientation of the edges and faces. Additionally, we introduce the following diagonalmatrices encoding the mesh information pertaining to MFD,

DeD = diag(|eDij |

), D∂D = diag (|∂Dkm|), DD = diag (|Dk|),

DeV = diag(|eVkm|

), D∂V = diag (|∂Vij |), DV = diag (|Vi|).

The matrix representations are derived from the component-wise definitions using theincidence matrices as the actions of the operators, and the diagonal matrices for theappropriate scaling. Thus,

gradD := D−1eDG, divD := D−1

V GTD∂V , curlD := D−1

∂DKDeD ,(2.5)

gradV := D−1eV GV , divV := D−1

D GTVD∂D, curlV := D−1

∂VKTDeV .(2.6)

With this construction, it is known that the mimetic operators are structure-preserving, i.e., curlDgradD = 0, curlV gradV = 0, divV curlD = 0, and divDcurlV = 0[8, 31]. Using these relationships, two exact sequences exist for MFD,

HDgradD−−−−→HD

curlD−−−→HVdivV−−−→ HV ,(2.7)

HVgradV−−−−→HV

curlV−−−→HDdivD−−−→ HD.(2.8)

Note again that the nature of the discretization method, when used for Maxwell’ssystem, enforces the PDE constraints strongly at the discrete level.

2.2. MFD for Maxwell’s Equations. Since the energy conservation propertyis important in electromagnetic applications, we consider the Crank–Nicolson scheme(a symplectic time integrator) with time-step τ and suitable initial conditions given by


appropriate interpolation to the dual meshes, p0D, E0

D, and B0V . The fully discretized

system becomes: find pnD ∈ HD, EnD ∈HD, and Bn

V ∈HV such that,

2

τBnV + curlDE

nD = gVB,(2.9)

2

τEnD − curlV B

nV + gradD p

nD = gDE ,(2.10)

2

τpnD + divDE

nD = gDp ,(2.11)

where the functions on the right-hand sides are given by,

gVB :=2

τBn−1V − curlDE

n−1D ,

gDE :=2

τEn−1D + curlV B

n−1V − gradD p

n−1D − (jnD + jn−1

D ),

gDp :=2

τpn−1D − divDE

n−1D .

The current density, jD ∈HD, is (jD)Dij = |∂V Dij |−1

∫∂Vij

j · nVij dx.

Remark 2.1. Recall that p(x, t) = 0 for all x ∈ Ω and t ≥ 0 with initial conditionp(x, 0) = 0. Therefore, (2.9)–(2.11) could be solved without including p and theanalysis that follows remains the same even without p. However, p is included todemonstrate the relationship between the grad, curl, and div spaces, which are chosensuch that the equations obey the mappings given by the sequences (2.7)–(2.8). Also,note that we could have put E and p on the Voronoi mesh, and B on the Delaunaymesh instead of the choice above. However, by putting the magnetic field on theVoronoi mesh, BV ∈ HV , guarantees divV BV = 0. Thus, the divergence of themagnetic field is constant zero on the Voronoi nodes which gives us a divergence-freemagnetic field on all Delaunay tetrahedra.

Using the definitions of the MFD operators, the linear system for the MFD scheme(2.9)–(2.11) is given by

(2.12)

2τ IeV D−1

∂D KDeD

−D−1∂V KT DeV

2τ IeD D−1

eD G−D−1

V GT D∂V2τ IV

︸︷︷︸

=:AMFD

BnV

EnD

pnD

=

gVBgDEgDp

.

3. Finite-Element Framework. Next, we consider a structure-preservingmixed FEM for the Maxwell system [4]. To approximate the inner-product terms onthe computational domain, we implement mass lumping, which results in diagonalmass matrices. This gives us FE blocks in terms of MFD mesh information, which isuseful when drawing connections in the next section.

Consider the differential operator, D, and Sobolev space,

H(D) := u ∈ L2(Ω),Du ∈ L2(Ω),

where D is grad, curl, or div. Let ‖·‖ and 〈·, ·〉 denote the L2 norm and inner product,respectively. Define the finite-dimensional function spaces with appropriate boundary


conditions, Hh,0(D). For the magnetic field, Bh ∈ Hh,0(div), we use the Raviart–Thomas (RT) element, the Nedelec element for the electric field Eh ∈ Hh,0(curl),and the Lagrange element for the auxiliary pressure ph ∈ Hh,0(grad). Defining Vh :=Hh,0(div)×Hh,0(curl)×Hh,0(grad), the FEM discretization for Maxwell’s equationsbecomes, find (Bn

h ,Enh , p

nh) ∈ Vh such that for all (Ch,Fh, qh) ∈ Vh,

2

τ〈Bn

h ,Ch〉+ 〈∇ ×Enh ,Ch〉 = (gB,Ch),(3.1)

2

τ〈En

h ,Fh〉 − 〈Bnh ,∇× Fh〉+ 〈∇pnh,Fh〉 = (gE ,Fh),(3.2)

2

τ〈pnh, qh〉 − 〈En

h ,∇qh〉 = (gp, qh),(3.3)

where the functionals on the right-hand side are given by,

(gB,Ch) =2

τ〈Bn−1

h ,Ch〉 − 〈∇ ×En−1h ,Ch〉,

(gE ,Fh) =2

τ〈En−1

h ,Fh〉 − 〈∇pn−1h ,Fh〉+ 〈Bn−1

h ,∇× Fh〉 − 〈jn + jn−1,Fh〉,

(gp, qh) =2

τ〈pn−1h , qh〉+ 〈En−1

h ,∇qh〉.

To write this as a linear system, we introduce discrete gradient and curl opera-tors. Let φgrad

i , φcurli , and φdiv

i be the bases of Hh,0(grad), Hh,0(curl), andHh,0(div), respectively. Let ηcurl

i be the degrees of freedom for the Nedelec space,and ηdiv

i for the RT space. Then, define the FE discrete gradient, GFE , and curl,KFE , in terms of the degrees of freedom,

GFEij := ηcurli

(∇φgrad

j

)=

1

|ei|

∫ei

∇φgradj · ti ds,(3.4)

KFEij := ηdivi

(∇× φcurl

j

)=

1

|fi|

∫fi

∇× φcurlj · ni dS.(3.5)

Note that the degrees of freedom are scaled by mesh data; the curl degrees of freedomare scaled by inverse edge lengths, and div degrees of freedom are scaled by inverseface areas. Thus, the FE linear system for (3.1)–(3.3) is given by

(3.6)

2τMB MBKFE

−(KFE

)TMB2τME MEGFE

−(GFE

)TME2τMp

︸︷︷︸

=:AFE

Bnh

Enh

pnh

=

gBgEgp

,

with mass matrices given by (Mp)ij = 〈φgradj , φgrad

i 〉, (ME)ij = 〈φcurlj ,φcurl

i 〉, and

(MB)ij = 〈φdivj ,φdiv

i 〉.

3.1. Mass Lumping. Now that we have linear systems for two discretizationmethods, our goal is to draw similarities between (2.12) and (3.6). Notice that the dis-crete differential operators are already in the same blocks; however, the MFD systemhas blocks with diagonal entries containing mesh information while the FEM systemhas mass matrices. Using ideas from [6, 7, 13, 30] to approximate the mass matri-ces with diagonal matrices, we implement mass-lumping schemes. The quadrature


schemes associated with the lumping result in a quantity that has physical signifi-cance, namely a volume unit for 3D meshes, and an area on the mesh for the 2D case.Therefore, we expect that mass lumping gives diagonal matrices where the diagonalrepresents some sort of volume (3D) or area (2D) in the dual mesh set-up. We see inthe next section that this is true for appropriate choices of quadrature weights, andthat these specific lumped matrices help draw more connections between the FE andMFD systems.

First, consider the Lagrange element mass matrix,Mp. For simplicity, assume wehave a regular primal (Delaunay) mesh for the finite-element grid, and an associateddual (Voronoi) mesh. The entries ofMp are integrals of the Lagrange basis functions,but can be rewritten in terms of a quadrature rule,

(Mp)ij = 〈φgradi , φgrad

j 〉 =

NV∑l=1

∫Dl

φgradi φgrad

j dx ≈NV∑l=1

d+1∑k=1

ωlk φgradi (xlk)φgrad

j (xlk),

where d is the dimension, the sum in l is over all of the elements, d+1 is the number ofnodes per element, and ωlk and xlk are the quadrature weights and nodes, respectively.Typically, Gaussian quadrature is used, but we manufacture a new rule such that thechoice of nodes and weights gives the approximation, Mp ≈ DV .

xD1xD2xD5

xD7xD6

xD4 xD3

xV1

xV2

xV3

xV4

xV5

xV6

Fig. 2: Two-dimensional subset of regular dual mesh (as in Figure 1). Delaunay meshin black solid lines, Voronoi mesh in red dashed line.

To illustrate this further, examine a subset of a regular dual mesh setup in 2D,shown in Figure 2. On the sub-mesh, we choose the quadrature points xlk to be theDelaunay nodes, xDm, and then determine what the appropriate weights should be.Consider two cases: i = j and i 6= j. When i 6= j, we have that all terms in the sumare zero, as φgrad

i (xDn ) = 0 if i 6= n and φgradj (xDn ) = 0 if j 6= n by the definition of the

finite-element basis functions and degrees of freedom. The only way to get a nonzeroterm in the sum is if i = j = n. When i = j, note that each term in the sum is onlynonzero when xlk = xDi . Consider the small mesh in Figure 2, and the case wheni = j = 1. Define the element enumeration using MFD notation, i.e., let D1 be theDelaunay element defined by Voronoi node xV1 , and let Dl be defined by xVl . Let k


enumerate the nodes in an element in increasing order. For example, for l = 1, wehave x11 = xD1 , x12 = xD2 and x13 = xD3 in Figure 2. The quadrature rule on thissub-mesh becomes,

NV∑l=1

d+1∑k=1

ωlk φgrad1 (xlk)φgrad

1 (xlk) = ω11φgrad1 (x11)φgrad

1 (x11) + ω21φgrad1 (x21)φgrad

1 (x21)

+ ω31φgrad1 (x31)φgrad


1 (x41)

+ ω51φgrad1 (x51)φgrad


1 (x61)

= ω11 + ω21 + ω31 + ω41 + ω51 + ω61.

To get the correct approximation by mass lumping, this sum must equal |V1|,the area of the Voronoi cell defined by the six Voronoi nodes. Therefore, we chooseω11 = |V1 ∩D1|, ω21 = |V1 ∩D2|, ..., ω61 = |V1 ∩D6|. Substituting in we see that,

NV∑l=1

d+1∑k=1

ωlk φgrad1 (xlk)φgrad

1 (xlk) = ω11 + ω21 + ω31 + ω41 + ω51 + ω61

= |V1 ∩D1|+ |V1 ∩D2|+ ...+ |V1 ∩D6|= |V1|.

The same argument can be applied more generally to the i = j case, where we wantωlk = |Dl ∩ Vi| when xlk = xDi . This gives us the approximation (Mp)ii ≈ |Vi|. Thesame idea applies to the 3D case, except we have four nodes in each element and theweights represent a volume instead of an area.

In general, for the Lagrange elements with mass matrixMp, we use the followingquadrature rule to lump the entries onto the diagonal. Summing over elements andnodes per element, we have for scalar functions u, v ∈ H(grad),

〈u, v〉Mp:=

NV∑l=1

d+1∑k=1

ωgradlk u(xlk)v(xlk), ‖u‖2Mp

:= 〈u, u〉Mp,

where the xlk are given by the nodes of the Delaunay mesh, and ωgradlk = |Dl∩Vi| when

xlk = xDi . Thus, we have thatMp ≈ Mp = DV , which is the standard mass-lumpingscheme for H(grad) [30].

Similarly for the RT elements, we introduce the following inner product, summingover elements and faces per element, for u,v ∈H(div),

〈u,v〉MB:=

NV∑l=1

d+1∑k=1

ωdivlk

(1

flk

∫flk

u · n dS)(

1

flk

∫flk

v · n dS),

‖u‖2MB:= 〈u,u〉MB

.

Using the same idea as the previous lumping scheme and following [7], we choose quad-rature weights to be ωdiv

lk =∣∣eVlm ∩Dl

∣∣ |∂Dlm| when flk = ∂Dlm, where ∂Dlm is theith face in the Delaunay mesh enumeration. Then, the mass matrix is approximatedby MB ≈ MB = D∂DDeV .


Finally, we examine the Nedelec element mass matrix, ME , by following [6]. Wehave the entries computed with the following quadrature rule, for u,v ∈H(curl),

〈u,v〉ME:=

ND∑l=1

d(d+1)2∑

k=1

ωcurllk

(1

elk

∫elk

u · t ds)(

1

elk

∫elk

v · t ds),

‖u‖2ME:= 〈u,u〉ME

,

where edge elk = eDmn is the ith edge in the Delaunay mesh enumeration, andelk = eDmn is contained in Delaunay tetrahedron Dl. Choosing weights to be ωcurl

lk =

|∂Vmn ∩Dl|∣∣eDmn∣∣ yields ME ≈ ME = D∂VDeD .

Remark 3.1. The mass-lumping schemes can be modified for non-constant ε(x)and µ(x) by taking a constant approximation of the coefficients on each element.With the piecewise constant approximation, the quadrature weights and sparsity ofthe mass-lumping schemes are unchanged.

Putting this all together we have the mass-lumped FE system as,

(3.7)

2τ MB MBKFE

−(KFE

)T MB2τ ME MEGFE

−(GFE

)T ME2τ Mp

︸︷︷︸

=:AFE

Bnh

Enh

pnh

=

gBgEgp

.

4. Connections between MFD and FEM. To study the well-posedness of themimetic discretization (2.9)–(2.11) and design efficient solvers for the resulting linearsystem, we draw connections to the FE scheme, noting that with mass lumping, thetwo systems have the same block structure. First, we rewrite the FE gradient andcurl operators in terms of the MFD incidence matrices as follows,

GFE = D−1eDG, KFE = D−1

∂DKDeD ,(4.1)

where we scale incident matrices on the mesh to be consistent with the definitions in(3.4)–(3.5). Applying a left scaling to the mass-lumping schemes, and substituting in(4.1) to (3.7), gives a new scaled FE linear system,

D−1∂DD

−1eV

D−1∂VD

−1eD

D−1V

AFE

︸︷︷︸=:ASFE

Bnh

Enh

pnh

=

D−1∂DD

−1eV

D−1∂VD

−1eD

D−1V

gBgEgp

.(4.2)

Substituting in Mp = DV , MB = D∂DDeV , and ME = D∂VDeD , we get ASFE =AMFD and recover exactly the MFD system in (2.12).

With the above equivalence, we apply a FE well-posedness proof to the mass-lumped FE system, thus obtaining the well-posedness of the mimetic system. Forsimplicity in dealing with the scaled system, we also introduce the function space,

Vh := Hh,0(div)× Hh,0(curl)× Hh,0(grad), which is Vh with basis functions scaledto reflect the left scaling in (4.2),


Hh,0(div) = spanφdivi , where φdiv

i =(D−1∂DD

−1eV

)iiφdivi ;

Hh,0(curl) = spanφcurli , where φcurl

i =(D−1∂VD

−1eD

)iiφcurli ;

Hh,0(grad) = spanφgradi , where φgrad

i =(D−1V

)iiφgradi .

Now, (4.2), and thus the MFD Maxwell System (2.12), can be written in variationalform (for simplicity, the subscripts indicating the time-step iteration and inclusion in

the FE space are excluded): find (B,E, p) ∈ Vh such that for all (C, F , q) ∈ Vh,

2

τ〈B, C〉MB

+ 〈∇ ×E, C〉MB= (gB, C),(4.3)

−〈B,∇× F 〉MB+

2

τ〈E, F 〉ME

+ 〈∇p, F 〉ME= (gE , F ),(4.4)

−〈E,∇q〉ME+

2

τ〈p, q〉Mp

= (gp, q).(4.5)

Note that MFD in this case can be considered as a Petrov-Galerkin FEM.Following the notation in [2], we introduce the bilinear form,

a(B,E, p; C, F , q) :=2

τ〈B, C〉MB

+ 〈∇ ×E, C〉MB− 〈B,∇× F 〉MB

(4.6)

+2

τ〈E, F 〉ME

+ 〈∇p, F 〉MB− 〈E,∇q〉ME

+2

τ〈p, q〉Mp

,

and the weighted norms

‖B‖2div :=2

τ‖B‖2MB

+ ‖∇ ·B‖2,(4.7)

‖E‖2curl :=2

τ‖E‖2ME

+τ

2‖∇ ×E‖2MB

,(4.8)

‖p‖2grad :=2

τ‖p‖2Mp

+τ

2‖∇p‖2ME

,(4.9)

||| (B,E, p) |||2 := ‖B‖2div + ‖E‖2curl + ‖p‖2grad.(4.10)

The following theorem shows that (4.3)–(4.5) is well-posed, and therefore (2.9)–(2.11)is well-posed.

Theorem 4.1. If gB ∈ (Hh,0(div))′, the MFD system (4.3)–(4.5) is well-posed,

namely, it satisfies the inf-sup condition,

(4.11) sup(C,F ,q)∈Vh

(C,F ,q)6=0

a(B,E, p; C, F , q)

|||(C, F , q )|||≥ 1

4||| (B,E, p) |||,

and is bounded,

(4.12) a(B,E, p; C, F , q) ≤ C|||B,E, p||| |||(C, F , q )|||.

Proof. Consider a variation of (4.6), where the term 〈∇ ·B,∇ · C〉 is added,

a(B,E, p; C, F , q) := a(B,E, p; C, F , q) + 〈∇ ·B,∇ · C〉.(4.13)

Since ∇ ·B = 0 for all t ≥ 0, (4.6) and (4.13) are equivalent. We proceed by showingthat (4.13) satisfies the inf-sup condition,

(4.14) sup0 6=(C,F ,q)∈Vh

a(B,E, p; C, F , q)

|||(C, F , q )|||≥ 1

4|||(B,E, p)|||,


and is bounded,

(4.15) a(B,E, p; C, F , q) ≤ C|||(B,E, p)||| |||(C, F , q )|||.

Let C = B+ τ2∇×E, F = E+ τ

2∇p, and q = p. Substituting into the bilinear form(4.13), using the fact that ∇ · ∇ ×E = 0 and ∇×∇p = 0, and simplifying, we have

a(B,E, p; C, F , q) =2

τ‖B‖2MB

+ ‖∇ ·B‖2 +2

τ‖E‖2ME

+τ

2‖∇ ×E‖2MB

+2

τ‖p‖2Mp

+τ

2‖∇p‖2ME

+ 〈∇ ×E,B〉MB+ 〈∇p,E〉ME

.

Using Young’s inequality to bound the cross terms, it follows that

a(B,E, p; C, F , q) ≥ 2

τ‖B‖2MB

+ ‖∇ ·B‖2 +2

τ‖E‖2ME

+τ

2‖∇ ×E‖2MB

+2

τ‖p‖2Mp

+τ

2‖∇p‖2ME

− τ

4‖∇ ×E‖2MB

− 1

τ‖B‖2MB

− 1

τ‖E‖2ME

− τ

4‖∇p‖2ME

=1

τ‖B‖2MB

+ ‖∇ ·B‖2 +1

τ‖E‖2ME

+τ

4‖∇ ×E‖2MB

+2

τ‖p‖2Mp

+τ

4‖∇p‖2ME

≥ 1

2|||(B,E, p)|||2.

Next, bound the norms of the test functions using the triangle inequality and Young’sinequality.

|||(C, F , q )|||2 = ‖B +τ

2∇×E‖2div + ‖E +

τ

2∇p‖2curl + ‖p‖2grad

≤ ‖B‖2div +τ2

4‖∇ ×E‖2div + τ‖B‖div‖∇ ×E‖div

+ ‖E‖2curl +τ2

4‖∇p‖2curl + τ‖E‖curl‖∇p‖curl + ‖p‖2grad

≤ ‖B‖2div +τ2

4‖∇ ×E‖2div + ‖B‖2div +

τ2

4‖∇ ×E‖2div

+‖E‖2curl +τ2

4‖∇p‖2curl + ‖E‖2curl +

τ2

4‖∇p‖2curl + ‖p‖2grad

≤ 4|||(B,E, p)|||.

Weak coercivity, (4.14), follows directly. To show boundedness, (4.15), apply Cauchy–Schwarz twice to (4.13). The well-posedness of bilinear form a defined in (4.13) followsdirectly from Babuska–Brezzi theory. Since a, (4.13), and the original bilinear forma, (4.6), are equivalent, the scaled, mass-lumped FE system, which is equivalent tothe MFD system, (4.3)–(4.5) is well-posed (similar arguments as in Lemma 1 andTheorem 8 of [18] give the result). This implies that (4.6) satisfies (4.11)–(4.12),which completes the proof.

5. Block Preconditioners based on Exact Block Factorization. One ofthe benefits of drawing connections between MFD and FEM, in addition to the abil-ity to show well-posedness of the MFD discretization, is that robust linear solversdeveloped for FEM [9, 10, 23, 24, 25] can now be applied to the MFD system. In this


work, we extend the ideas from [2, 24] to the MFD system and develop robust pre-conditioners based on block factorization, exploiting the structure-preserving natureof the discretization.

Good block preconditioners are often based on Schur complements and their ap-proximations, and the accuracy of the approximations greatly influences the perfor-mance of the preconditioner. However, the structure-preserving discretization allowsfor Schur complements to be computed exactly, and the exact sequence of the discretespaces yields sparse Schur complements that are used directly without approximation.

To more clearly exploit the structure-preserving nature of the MFD discretization,we re-write the blocks of (2.12) back in the MFD operator notation given by (2.5)–(2.6),

(5.1) AMFD x = b ⇐⇒

2τ IeV curlD

−curlV2τ IeD gradD−divD

2τ IV

Bn

V

EnD

pnD

=

gVBgDEgDp

.Recall that the structure-preserving discretization enforces the properties of the gra-dient, curl, and divergence (curl grad = 0 and div curl = 0) on the discrete level ascurlDgradD = 0, curlV gradV = 0, divV curlD = 0, and divDcurlV = 0. Exploitingthese properties gives the exact block factorization of (2.12),(5.2)

AMFD =

IeV

− τ2 curlV IeD

− τ2 divD IV

︸︷︷︸

L

2τ IeV

SESp

︸︷︷︸

S

IeVτ2 curlD

IeDτ2 gradDIV

︸︷︷︸

U

,

with the Schur complements computed exactly as,

SE =τ

2curlV curlD +

2

τIeD , Sp =

τ

2divDgradD +

2

τIV .

For the remainder of the paper, we drop the subscript notation and just representAMFD by A.

Several block preconditioners can be designed from the exact factorization, (5.2).A natural choice of preconditioner is S−1. However, this involves computing theinverse of the Schur complements, making this choice impractical. To rectify this,replace the Schur complements, SE and Sp, with good preconditioners, QE and Qp.For example, an HX-preconditioner [17] can be used for QE and a standard multigridpreconditioner for Qp. Note that the top left entry in S is a scaled identity matrix,so no spectrally-equivalent approximation is needed. For the remaining two blocks,we assume that

c1,E ≤ λ(QESE) ≤ c2,E ,(5.3)

c1,p ≤ λ(QpSp) ≤ c2,p.(5.4)

This implies that for Q = diag((

2τ IeV

)−1,QE ,Qp

), we have

c1 ≤ λ(QS) ≤ c2,

where c1 = min(c1,E , c1,p) and c2 = max(c2,E , c2,p).


From this factorization, we consider three different block preconditioners,

(5.5) XLS := QL−1, XSU := U−1Q, XLSU := U−1QL−1,

where L−1 and U−1 can be computed directly,

L−1 =

IeV

τ2 curlV IeD

τ2 divD IV

, U−1 =

IeV − τ2 curlD

IeD − τ2 gradDIV

.In the following theorem, we prove that these preconditioners are robust with respectto the discretization parameters. By bounding the eigenvalues for the preconditionedsystem, we guarantee good performance of GMRES. Note that the following proofcan be done to show that the constants c1 and c2 are also independent of the PDEparameters, ε and µ, but for simplicity we only consider the case where ε = µ = 1.Otherwise, the identity matrices in the diagonal matrix of the decomposition wouldbe scaled by the PDE parameter values.

Theorem 5.1. Let XLS , XSU , and XLSU be defined by (5.5) and assume thespectral equivalent properties (5.3)–(5.4) hold. Then,

(5.6) λ (XLSA) ∈ [c1, c2], λ (XSUA) ∈ [c1, c2], λ (XLSUA) ∈ [c1, c2],

where c1 = min (c1,E , c1,p) and c2 = max (c2,E , c2,p) are constants that do not dependon discretization parameters, h and τ .

Proof. First, consider XLSA,

XLSA = QL−1LSU = QSU =

IeVτ2 curlD

QESE QEgradDQpSp

.The eigenvalues λ (XLSA) are determined by the eigenvalues of the diagonal blockssince XLSA is block upper triangular. The first block is an identity matrix, whoseeigenvalues are all ones. For the other two blocks, we use (5.3)–(5.4) to bound theeigenvalues and, overall, have λ (XLSA) ∈ [c1, c2].

To bound the eigenvalues of XSUA, consider the eigenvalue problem,

XSUAx = λx ⇐⇒ Ax = λX−1SUx.

By substituting in the decompositions of A and XSU , it follows that,

LSUx = λQ−1Ux.

Let y = Ux and left multiply by Q. Then, QLSy = λy, where

QLS =

IeV

−QEcurlV QESE−QpdivD QpSp

.SinceQLS is block lower triangular, again the eigenvalues only depend on the diagonalblocks. Thus, by (5.3)–(5.4), λ (XSUA) ∈ [c1, c2].


Finally, consider the preconditioned system

XLSUAx = U−1QL−1LSUx = U−1QSUx = λx.

Left multiplying by U and letting y = Ux yields,

QSy = λy,

where

QS =

IeV

QESEQpSp

.By the same reasoning as the other two cases, we conclude λ (XLSUA) ∈ [c1, c2].

5.1. Preservation of the Divergence-Free Magnetic Field. The goal ofusing a structure-preserving discretization for the Maxwell system is to enforce thePDE constraints on the discrete level. However, even if the discretization holds, theapproximate solve for each time step of a linear solver can destroy these proper-ties. The following theorem ensures that at each iteration of the linear solver, thedivergence-free condition for B is preserved.

Theorem 5.2. Let x0 =(B0V ,E

0D, p

0D

)Tbe the initial guess for the MFD system

satisfying divV B0V = 0, and let b =

(gVB, g

DE , g

Dp

)Tbe the MFD right-hand side sat-

isfying divV gVB = 0. Then all iterations, xk =

(BkV ,E

kD, p

kD

)T, of the preconditioned

GMRES method satisfy divV BkV = 0, where X is any of the preconditioners defined

in (5.5).

Proof. Define the Krylov subspace,

Kk(XA, r0) = spanr0,XAr0, (XA)2r0, ..., (XA)

kr0

with r0 =(r0B, r

0E , r

0p

)T:= X

(b−Ax0

)such that for each iteration,

(5.7) xk ∈ x0 +Kk(XA, r0

).

By the assumption of the divergence-free initial data, it follows that divV r0B = 0.

Next, let vm =(vmB ,v

mE ,v

mp

)T:= (XA)

mr0 for m = 0, 1, ..., k − 1. Since vm =

XAvm−1, we compute vmB for each of the preconditioners in (5.5). Setting X = XLS ,

vmB =

(2

τIeV

)−1(2

τvm−1B + curlDv

m−1E

)= vm−1

B +τ

2curlDv

m−1E .

For X = XSU ,

vmB = vm−1B +

τ

2curlD

(vm−1E +QEcurlV v

m−1B − 2

τQEv

m−1E −QEgradDv

m−1p

),

and for X = XLSU ,

vmB = vm−1B

+ curlD(τ

2vm−1E − τ2

4QEcurlV curlDv

m−1E −QEv

m−1E − τ

2QEgradDv

m−1p

).


Applying divV to vmB for each preconditioner above, all terms with curlD in frontare zero since divV curlD = 0. Then, divV v

mB = 0 if divV v

m−1B = 0. By an inductive

argument, since divV r0B = 0, we have that divV v

mB = 0. By (5.7) and the definition

of vm, xk is a linear combination of vm, m = 0, 1, ..., k − 1 . This implies that BkV is

a linear combination of vmB . Since divV vmB = 0, then divVB

kV = 0 for all k.

6. Numerical Results. To demonstrate the theoretical results presented inthe previous sections, consider the following test problem with essential Dirichletboundary conditions,

E(x, t) =1

πe−t

− cos(πx1) sin(πx2) sin(πx3)

sin(πx1) cos(πx2) sin(πx3)

0

,(6.1)

B(x, t) = e−t

− sin(πx1) cos(πx2) cos(πx3)

− cos(πx1) sin(πx2) cos(πx3)

2 cos(πx1) cos(πx2) sin(πx3)

,(6.2)

p(x, t) = 0,(6.3)

j(x, t) = −e−t

(1

π+ 3π

) cos(πx1) sin(πx2) sin(πx3)

− sin(πx1) cos(πx2) sin(πx3)

0

.(6.4)

While the analysis presented in this paper holds for a general Voronoi mesh, forsimplicity, we consider a non-degenerate mesh, where the Voronoi points do not lieon the boundary or outside of the corresponding Delaunay tetrahedra. To get a non-degenerate Voronoi mesh, we design a Delaunay triangulation for the FE domain thatconsists of a cube with a rectangular pyramid on each face (see Figure 3). The pyra-mids are defined by the faces of the unit cube and the points

(− 1

2 ,12 ,

12

),(

32 ,

12 ,

12

),(

12 ,−

12 ,

12

),(

12 ,

32 ,

12

),(

12 ,

12 ,−

12

),(

12 ,

12 ,

32

). Uniform refinement is used to get more re-

solved meshes and Table 1 lists the geometric information for the different resolutionsconsidered. Numerical experiments are implemented in the HAZmath package [1]written by the authors. All timed numerical results are done using a workstationwith an 8-core 3-GHz Intel Xeon Sandy Bridge CPU and 256 GB of RAM.

Fig. 3: Left: Delaunay mesh with h = 1; Right: cross section at z = 1/2


h Vertices Edges Faces DoF

Mesh 1 1/4 369 2,096 3,264 5,729

Mesh 2 1/8 2,465 15,520 25,344 43,329

Mesh 3 1/16 17,985 119,360 199,680 337,025

Mesh 4 1/32 137,345 936,064 1,585,152 2,658,561

Mesh 5 1/64 1,073,409 7,414,016 12,632,064 21,119,489

Table 1: Geometric information for the Delaunay meshes.

To demonstrate the equivalence of the two discretizations, MFD and FEM areimplemented and solved with XLSU -preconditioned FGMRES with restart after 100iterations to a relative residual tolerance of 10−8. We expect FE convergence rateswith respect to mesh size for both. More precisely, with L2 norm in space and L∞

norm in time, E and B modeled with lowest order Nedelec and Raviart-Thomaselements, respectively, are expected to converge with O(h+ τ2), where h is the meshpartitioning and τ is the time-step size. Results for p are excluded as it is just anauxiliary variable with no physical relevance to the problem. By choosing a small timestep and few time iterations (τ = .0125 for 8 time steps, i.e., final time t = 0.1), thespatial error dominates. This is confirmed in Figure 4, where the spatial convergenceof the MFD scheme is identical to the FEM one.

Fig. 4: L∞([0, t];L2(Ω)

)error for FEM (left) and MFD (right) for τ = 0.0125 after

8 time steps (t = 0.1) solving with XLSU preconditioner and FGMRES.

Next, performance comparisons are made between the MFD and FE methodsusing the preconditioners developed in Section 5 for MFD and in [2] for FEM. Againusing test problem (6.1)–(6.4) and meshes given by Table 1, we test the precondition-ers defined in (5.5) for robustness with respect to time step, τ , and mesh size, h. Thediagonal blocks of the preconditioners are solved inexactly by preconditioned GMRESwith a relative residual reduction set at 10−2 for the outer iteration. Flexible GMRESis used as the outer iteration with a relative residual stopping criteria of 10−8, withrestart after 100 iterations. Iteration counts for both MFD and FEM with precon-ditioners XLS , XSU , and XLSU are reported in Tables 2 and 3. In all experiments,iteration counts are averaged over the number of timesteps needed to reach t = 1.The spectral equivalent approximations used in Q are an HX preconditioner [17, 20]


for SE , and a standard algebraic multigrid method for Sp.The results show little variation in iteration count with varying parameters, indi-

cating that the block preconditioners based on exact block factorization are effectiveand robust with respect to mesh size, h, and time step size, τ , for both MFD andFEM as expected. Furthermore, MFD has comparable iteration counts to FE.

XLS XSU XLSU

τ

h 14

18

116

132

164

14

18

116

132

164

14

18

116

132

164

0.2 5 5 5 6 8 5 5 5 6 6 4 3 4 4 4

0.1 3 4 5 5 6 4 5 5 5 6 2 3 3 4 4

0.05 4 3 4 4 5 4 4 5 5 5 3 2 3 3 3

0.025 3 4 3 4 4 3 4 4 5 5 2 3 2 3 3

0.0125 2 3 4 3 3 3 3 4 4 4 2 2 3 2 3

Table 2: Iteration counts for the block preconditioners based on block factorizationfor MFD. Left: block lower triangular, XLS ; Center: block upper triangular, XSU ;Right: symmetric, XLSU .

XLS XSU XLSU

τ

h 14

18

116

132

164

14

18

116

132

164

14

18

116

132

164

0.2 4 4 4 4 4 5 5 5 5 4 3 3 3 3 3

0.1 5 4 4 4 4 5 5 5 5 4 3 3 3 3 3

0.05 4 4 4 4 3 5 4 4 4 4 3 2 3 3 3

0.025 3 3 3 3 3 4 4 3 3 2 3 2 2 2 2

0.0125 3 3 3 3 3 3 3 4 3 3 3 2 2 2 3

Table 3: Iteration counts for the block preconditioners based on block factorizationfor FEM. Left: block lower triangular, XLS ; Center: block upper triangular, XSU ;Right: symmetric, XLSU .

To further compare the two methods, CPU solve time per time iteration is exam-ined for fixed time-step size, as well as the time scaling of the solve time per time stepfor the three preconditioned FGMRES solvers, both with respect to mesh refinement.In Table 4, the average solve time over ten iterations with fixed τ is reported. Wesee that as the number of degrees of freedom increases, the FE method beats MFD,especially for XLS and XSU . This is further demonstrated in Figure 5, where a side-by-side comparison of the three preconditioners for MFD and FEM is given. This timedisparity is likely given by the fact that MFD takes slightly more GMRES iterationsper solve, which could be due to the fact that the MFD system is not symmetric whileFEM is.


h 14

18

116

132

164

MFD

XLS 1.32×10−2 1.06×10−1 9.95×10−1 1.31× 101 1.68× 102

XSU 1.48×10−2 1.12×10−1 1.10× 100 1.41× 101 1.96× 102

XLSU 1.06×10−2 8.32×10−2 8.15×10−1 1.12× 101 1.41× 102

FEM

XLS 2.05×10−2 1.08×10−1 8.84×10−1 9.62× 100 1.09× 102

XSU 2.45×10−2 1.24×10−1 1.02× 100 1.19× 101 1.17× 102

XLSU 1.79×10−2 9.17×10−2 7.57×10−1 8.04× 100 1.06× 102

Table 4: Average CPU solve time per time iteration over ten time steps with τ = 0.1for MFD and FEM with all preconditioners.

Fig. 5: Comparison of solve time averaged over ten time steps of τ = 0.1 for MFDand FEM for fixed mesh size h = 1

64 . The number of FGMRES iterations is givenabove each bar.

Finally, it is expected that the solve time scales O (N log(N)), where N is thenumber of degrees of freedom. Figure 6 verifies this, where both FE and MFD resultsfollow the trend of the reference line for all three preconditioners.

Based on the results presented, we conclude that all three preconditioners arerobust and effective for solving the MFD system, and give comparable results to theFE method. In terms of both solve time and iteration counts, XLSU is the bestpreconditioner for both methods, which was also concluded in [2] for the FE method.

7. Conclusions. By examining MFD in a FE framework, we are able to ex-ploit the FEM tools and theory to strengthen the MFD method for Maxwell. Fromthis equivalence of the two methods, well-posedness of the MFD system is provenwith Babuska-Brezzi theory. Numerical results demonstrate that the FE convergencetheory is recoverable in the MFD implementation. Furthermore, in showing well-posedness of the Maxwell MFD discretization and the connection to the FEM, robustblock preconditioners developed for the FEM in [2] are adopted for solving the MFD


Fig. 6: Time scaling of solve time averaged over ten time steps of τ = 0.1 for FEM(left)and MFD(right), where N is the number of degrees of freedom.

linear system efficiently. When using structure-preserving discretizations, the goal isto enforce the PDE constraints, particularly a divergence-free magnetic field for Max-well’s equations, at all time steps and all solve iterations. The block preconditionersfor GMRES developed here are shown to be robust and guarantee all properties ofthe discretization at each time step.

All of the results in this paper apply to lowest–order FE and MFD methods. Whilehigher–order MFD methods exist [14, 21, 22], connections to higher–order FEM areunclear and require further investigation. If such relationships are found, the analysispresented here should be valid. Additionally, a priori error estimates for MFD forMaxwell could be derived in a fashion similar to the techniques used for Darcy flowin [13]. Furthermore, now that a FE framework for MFD has been developed forthe full Maxwell system, other electromagnetic applications can be explored such asMHD, time-harmonic Maxwell, or H(curl) and H(div) problems, in general. Thestructure-preserving nature of the MFD discretization opens the door for many morephysical applications with PDE constraints to be explored.

Acknowledgments. The work of JHA and XH was partially funded by NationalScience Foundation grant DMS-1620063. The work of LTZ was supported in part byNSF DMS-1720114 and DMS-1819157.

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