A SECOND ORDER DISCRETIZATION OF MAXWELL’SEQUATIONS IN THE QUASI-STATIC REGIME ON OCTREE GRIDS

LIOR HORESH∗ AND ELDAD HABER†

Abstract. In this study we consider adaptive mesh refinement for the solution of Maxwell’sequations in the quasi-static or diffusion regime. We propose a new finite volume OcTree discretiza-tion for the problem and show how to construct second order stencils on Yee grids, extending theknown first order discretization stencils. We then develop an effective preconditioner to the problem.We show that our preconditioner performs well for discontinuous conductivities as well as for a widerange of frequencies.

Key words. OcTree, adaptive mesh refinement, 2nd order, Maxwell solver, multi-grid, adjointcurl, quasi-static

AMS subject classifications. 65D05, 65D25, 65F10, 65F50, 65G30, 65M06, 65M12, 65M15,65M55, 65M50,

1. Introduction. Nowadays, simulation-based modeling of low-frequency elec-tromagnetic fields is becoming an essential tool in fields such as geophysics, medicalimaging, magnetohydrodynamics, electric-machine design, magnetic actuator designand non-destructive eddy-current testing simulations (see for example [24, 23, 36, 60,61] and reference within). Modeling accuracy can broadly be attributed to two mainfactors: geometry representation and fields variability. First, the geometry of thedomain of interest should be represented with a sufficient level of detail. Second, asufficiently fine discretization is required to capture variations of fast changing fields.Our interest stems from problems in geophysics and medical physics [18, 61] wherethe domains and boundary conditions are relatively simple, but the structure of thecoefficients (conductivity and permittivity in the context of Maxwell’s equations) canbe complicated.

Initially, most finite difference algorithms for the solution of Maxwell’s equationsused Yee’s method [63, 51, 38, 18], that is, the equations were discretized using a finitevolume/difference method on staggered grids. These methods maintained favorablemimetic properties, however, they were originally designed to deal with regular grids.A generalization of such methods for unstructured grids was proposed in [7, 5, 9].Such generalization is closely related to the mixed finite element discretization [36].

While unstructured grids offer great flexibility in terms of modeling complex ge-ometries and accuracy, this choice also endures a number of drawbacks. First, meshingis a non-trivial task and typically requires sophisticated meshing tools [55]. This be-comes even a greater difficulty when geometrical multi-grid schemes are sought, asan hierarchy of meshes is required. Second, in the lack of sufficient care during themesh generation procedure, matrices obtained using unstructured mesh are prune tobe highly ill-conditioned, even for well-conditioned problems [49, 3, 48]. Third, the re-sulting matrices are unstructured, thus, the application of optimal solution algorithmssuch as algebraic multi-grid becomes difficult [23, 43, 58].

Finally, while the above shortcomings are innate, another complication associatedwith unstructured mesh discretization arises while discretizing vector fields using stag-gered grids (or Nedelec elements). Since edges are generally misaligned with the axes

∗Business Analytics and Mathematical Sciences, IBM TJ Watson Research Center, YorktownHeights, New York, 10598 (lhoresh@us.ibm.com).†Mathematics, University of British Colombia, Vancouver, Canada, V6T1Z2

1

2 L. HORESH AND E. HABER

of the computation, non-trivial interpolation is essential for the evaluation of the fieldsat any given point [25].

The above difficulties have motivated us to explore the use of local mesh re-finement strategies over semi-structured orthogonal grids, sometimes referred to asOcTree discretization.

The spatial domain is represented by cubical cells, where the volume representedby each OcTree cell may be subdivided into eight octants. OcTree structures (e.g.,Figure 2.1(a)) bears some resemblance to another space-partitioning data structurecalled KD-Trees. However, the latter introduces splits along a dimension whereasOcTrees split around a point. Octree structures among other hierarchical data struc-tures that rely on recursive decomposition, have been used extensively in computergraphics, computer aided design, computer vision and cartography [44]. Their utilityin the context of PDEs has been proposed, for example in [13, 12, 33, 31, 4, 45, 50].Literature review regarding locally refined meshes, reveals extensive body of workdedicated to locally adaptive mesh refinement for Poisson-like equations with varyingcoefficients in either first or second order formulation [13, 12, 31, 33, 56].

Discussion regarding the solution of Maxwell’s equations on locally refined Carte-sian meshes, also known in the literature by the term “sub-gridding”, has been pre-sented in a number of studies, mainly in the context of Finite Difference Time Domain(FDTD) formulations [30, 64, 52, 8, 39, 59, 41, 19]. Some studies have explored inter-esting notions, such as non-binary division [64, 8] or offset grid collocation [39], yet,in common, all were based on extensions of Yee’s method to locally refined mesh andlimited to first-order formulations.

Indeed, locally refined Cartesian meshes may be less optimal compared to theirunstructured counterparts in terms of representation efficiency, that is, representationof complex geometries by a small number of variables. Nevertheless, they do offermany favorable properties. In particular, meshing and mesh refinement are almosttrivial and the structure of the matrices is “predictable”, since, only a small numberof stencils participate in the discretization process. The main advantage in havinga limited number of differentiation stencils is the consequent control one gains uponconditioning. It is straightforward to numerically verify that none of the stencilslead to ill-conditioning. This is in contrast to unstructured meshes where a singlesharp-angled tetrahedron can lead to ill-conditioning of the resulting discrete system.Furthermore, it was shown in [19] that derivation of a geometric multi-grid algorithmfor such a discretization can be done in a straightforward manner, and numericalexperiments indicate that such multi-grid algorithms converge even for high contrastcoefficients [19].

The major difficulty in discretizing PDEs on locally refined meshes is derivingdiscrete differential operators around the interface of varying cell sizes. This no-torious task has been investigated extensively for the context of Poisson equation[13, 26, 33, 35, 4], yet, it introduces even graver challenges for staggered grid dis-cretization of PDEs involving curl operators1. Specifically, in order to obtain ac-curate discretizations for such PDEs, special care must be devoted to address theso-called “hanging faces” and ”hanging edges” of uneven cell sizes, which are the kinof the “hanging nodes” in the context of Poisson equation. For the latter, a number ofmethods to approximate Poisson equation (in either first or second order form) aroundcell-interface [13, 12, 31, 33] exists. Our goal here is to develop similar techniques forMaxwell’s equations. It is important to note that this task was not performed by other

1Further details on that regard can be found in section 2.5)

2nd ORDER OCTREE DISCRETIZATION 3

papers that discuss sub-gridding for Maxwell’s equations. Several numerical studiesindicate that low accuracy, e.g. O(h), may not impair the overall accuracy, as long asthe discretization error lies on a small manifold [54, 34, 28, 62, 37, 26, 35]. Nonethe-less, it is possible to verify that the finite difference stencil suggested by sub-griddingmethods have O(1) truncation accuracy that results in O(h) approximation to thesolution. Our goal is therefore to understand the order of accuracy at the hangingedges, improve it and generate numerical solvers that can deal with our discretization.

In this paper, we therefore study adaptive mesh refinement discretization strate-gies which generalizes ideas that were developed for nodal discretizations to staggeredgrids. Our goal is to explore and experiment with methods that allow for secondorder accuracy even around uneven cell interfaces. In particular, we derive secondorder discretization for quasi-static Maxwell’s equations around an interface. Similarto the discretization of Poisson equation, the construction of higher order stencilsaround an uneven interface entails a loss of symmetry of the resulting system. In thecontext of Maxwell’s equations, one also looses some of the mimetic properties arounduneven interfaces. This may raise some difficulties when solving the resulting linearsystem. We explore this issue and show how to obtain efficient linear algebra solversfor different discretizations of the system.

The remainder of the paper is structured as follows. In Section 2 we review thediscretization of Maxwell’s equations using a locally refined mesh. In Section 3 wediscuss a new discretization technique around hanging faces. We show that in orderto obtain a second order accuracy six local topologies need to be considered. InSection 4 we discuss the solution of the discrete system of equations and discuss theapplication of a multi-grid preconditioner. In Section 5 we elucidate the results ofsome of the numerical experiments we conducted and finally in Section 6 we concludeand summarize the study.

2. OcTree discretization of Maxwell’s equations. In this section we presentthe discretization of Maxwell’s equations on OcTree mesh. We begin with a reviewof the discretization of the div grad and curl on staggered grid as presented in[8, 19, 39] and discuss the difficulties in extending this discretization to second orderaccuracy. We then present a framework that allows the construction of a second orderdiscretization of Maxwell’s equations and discuss it in detail.

2.1. Maxwell’s equations. Maxwell’s equations in the frequency domain canbe written as

∇× ~E = −iωµ ~H (2.1a)

∇× ~H = γ ~E + ~q (2.1b)

n× ~H = 0 (2.1c)

where ~E and ~H are the electric and magnetic fields, µ is the magnetic permittivity,γ ≡ σ+ iωε is the complex conductivity where σ is the conductivity, ω is the angularfrequency and ε = ε0εr is the dielectric permittivity, q stands for an external currentdensity source. Here we chose boundary conditions on the tangential components of~H but other appropriate boundary conditions can equally be prescribed.

For our applications we consider the quasi-static case where ωε� σ and thus thesystem can be viewed as a diffusion equation [61, 60]. This case is typically valid forgeophysical and medical applications; It implies that the dimensions of the domain ofinterest are substantially smaller than the size of a single wavelength [11].

4 L. HORESH AND E. HABER

There are three main difficulties in solving the system for geophysical or medicalapplications. First, conductivity σ varies over several orders of magnitude and hencethe PDEs involve non-smooth coefficients which connote non-smooth solutions. Sec-ond, due to the nontrivial null-space of the curl operator the resulting linear systemsare highly ill-conditioned. Finally, the fields may vary rapidly in the vicinity of sourcesand become rather smooth afar, requiring very high resolution at some regions of thedomain.

2.2. OcTree discretization of differential operators.

(a)

S(:, :, 1) =

(2 0 1 10 0 1 1

)

S(:, :, 2) =

(0 0 1 10 0 1 1

)

(b)

Fig. 2.1. (a) OcTree and (b) and its representation as a 2×4×2 (sparse) array. The notationS(:, :, i) implies the ith plane in the 3D array S.

2.2.1. OcTree structure. To discretize Maxwell’s equations we consider a fineunderlying orthogonal mesh of size 2m1 × 2m2 × 2m3 with grid size h. Our 3D grid iscomposed of m square cells of different sizes. The length size of each cell may carrypowers of 2 multiplied by h. To make the data structure easier and the discretizationmore accurate we employ a graded tree structure [42], that is only a factor of 2 inlength size between adjacent cells is permitted. The data is then stored as a sparsearray. The size of each cell is stored in the upper left corner of the array.

This data structure allows us to locate neighbors quickly, an operation that furtheron unveils as fundamental in the discretization process. This data structure is closelyrelated to the one suggested in [22]. An example of a small 3D OcTree grid is displayedin Figure 2.1.

2.3. Discretization of the div and grad. We are interested in the solutionof Maxwell’s equations (2.1) where effectively the div and grad operators do notappear explicitly. Nonetheless, as we see next, reasoning the difficulties here helps inperceiving the intricacies introduced by the discretization of the curl and its adjoint.We will also use the grad operator later for the null space of the curl.

We shall first consider the 2D case for simplicity. We use the usual flux-balanceapproach in order to discretize the divergence operator. Consider a 2D cell as shownin Figure 2.2. We discretize the flux density vector, ~J = (Jx, Jy)> on the faces ofthe cell and using Gauss theorem we formulate the discrete divergence of cellj with

2nd ORDER OCTREE DISCRETIZATION 5

φ1

φ2

φ3

φGcellj

jx1

jx2

jx3

jy1

jy2

Fig. 2.2. Discretization of the divergence

volume Vj

1

Vj

∫cellj

∇ · ~J dV =1

Vj

∫faces cellj

~J · d~S ≈ 1

2Vj(2jx1

− jx2− jx3

+ 2(jy1− jy2

)).

It is easy to verify that this standard midpoint discretization of the integral is secondorder accurate. Such mass-balance approximation can be formed for each cell in ourgrid, resulting in a discretization of the div operator.

The adjoint operator to the div, that is the gradient, maps from cell-centers tofaces. The discretization of the gradient, is more complicated, since no elementarydiscretization that is second order exists. Consequently, various formulations havebeen proposed for the definition of the discrete gradient operator.

Let us consider the gradient at the point x2 in Figure 2.2. The use of a two-pointformula was proposed in [32, 13] and reads

(φx)x2≈ 1

h(φ2 − φ1) (2.2)

It is easy to show that this stencil leads to a gradient matrix that is a scaled transposeof the divergence and therefore yields matrices that are symmetric positive and semi-definite. It was claimed in [32] that this form is an O(1) approximation to the gradientbut as we observe next, this is not really the case. Other, higher order approximationscan be used [27, 13]. In particular, one may consider the formula

(φx)x2≈ 1

3h(φ2 + φ3 − 2φ1) (2.3)

which seems like an O(h) approximation, although as we show next it is actuallyO(h2). The benefit of accuracy obtained by this approximation is somewhat hamperedby the loss of symmetry. Indeed, the approximation (2.3) is not a scaled transposeof the divergence. Nonetheless, as proved in [27] combining the divergence and thegradient obtained by (2.3) yields a non-symmetric M-matrix which still maintainfavorable properties.

6 L. HORESH AND E. HABER

We shall now re-examine the approximations (2.2) and (2.3). It is evident thattheir truncation error is indeed O(1) and O(h), respectively. This may lead to theconclusion that the discretization error in the solution is expected to be of a similarorder. However, as observed in [32] this is not the case and, even though the trun-cation error seem to be O(1) the solution error is actually O(h). In our numericalexperiments we have also found that the solution error that corresponded to (2.3) wasof a second order accuracy.

To understand this seemingly puzzling phenomenon one must interpret the linearsystem and its associated truncation error carefully. Rather than using Taylor’s seriesaround x2 we introduce a ghost point φG (Figure 2.2). It is obvious that

φx =1

h(φG − φ1) +O(h2) (2.4)

The difficulty is that the value of φG is unknown and therefore interpolation is needed.In general, one can write

φG =∑j

wjφj +O(hp) (2.5)

where wj are weights and p is the order of interpolation. For, example, one can use afirst order interpolation

φG = φ1 +O(h)

or, a second order interpolation which can be obtained by passing a plane throughx1, x2 and x3

φG = φ1 +1

3(φ2 + φ3 − 2φ1) +O(h2)

Now consider the system of equations given by the two equations (2.4) and (2.5).It is obvious that the truncation error for this system is min(2, p). The formulas (2.2)and (2.3) are obtained by elimination of the ghost point, however, they do not alterthe overall solution error obtained by the discretization. Since the PDE is discretizedto O(hmin(2,p)) we should expect the error to conform in such rate. This in factexplains the O(h) convergence observed for the “so-called O(1)” truncation error andthe O(h2) convergence observed for the “so-called O(h)” truncation error.

Let us now generalize this formulation. We denote the regular grid points by uI ,and respectively the ghost points by uG. The resulting discretization can be castedas a linear system for both sets of points(

AII AIG

AGI −IGG

)(uIuG

)=

(bI +O(hs)O(hp)

)(2.6)

where O(hs) is the error of the standard stencil operator which poised upon the hybridsupport uI , uG

2.The system AGIuI − uG = O(hp) comprises the interpolation conditions. Unfor-

tunately, in most cases AGI 6= A>IG and thereby symmetry is lost. Nonetheless, asasserted above, this system grants discretization accuracy of the order of min(s, p). In

2In the most common case the stencil is the standard second order finite difference stencil, yet,the extension to any other order is trivial

2nd ORDER OCTREE DISCRETIZATION 7

some cases, it is possible to show that if a first order interpolation is employed (thatis piecewise constant) then it is possible to obtain a symmetric system. Some mayadvocate that symmetry preservation is important and prefer to sacrifice the orderof discretization merely to obtain a symmetric positive definite system, which can beeasily handled by standard sparse solvers. In our view symmetry preservation is oflesser importance, if the matrix maintains some properties (such as M-matricity) andappropriate linear algebra is practiced.

2.4. The discretization of the nodal gradient. Although the discrete nodalgradient is not mentioned explicitly in the discretization of Maxwell’s equations, itsrole will become apparent further on in the solution process, and therefore, its deriva-tion is described briefly here. The discrete nodal gradient operator is a natural opera-tor that maps nodes into edges. The grad is obtained by differencing adjacent nodesand division by the edge length. It is possible to show [19] that if C is the discretiza-tion of the curl, as derived in the following subsection, and Gn is the discretizationof the nodal grad then the following mimetic property holds CGn = 0.

2.5. Discretization of the curl. Equipped with the insights acquired for thediscretization of the div and grad operators we shall now derive a discretization forthe curl and its adjoint.

Similar to the discretization of the div we use integral identities in order todiscretize the curl. We note from Stokes theorem that∫

cell face

∇× ~H · d~S =

∮cell edges

~H · d~

where d~S is a surface area normal to the field and d~ is a closed loop. Consider theupper face plotted in Figure 2.3. A straightforward second order discretization reads

1

Fj

∫facej

∇× ~H · d~S ≈ 1

2Fj(2hx1

− hx2− hx3

+ 2 (hy1− hy2

))

where as illustrated above Fj is the area of the face. Similar to the above discretizationwe integrate over every cell-face in our grid to obtain the discretization of the curl.Note that the discrete curl is a mapping from cell edges to cell faces.

Similar to the div, the curl is naturally discretized from edges to cells. Likewisethe div-grad case, difficulties emerge when discretizing the adjoint of the curl. Notethat the adjoint of the curl maps cell faces into cell edges. One option is to use a scaledtranspose of the curl which indeed maps between the correct entities. This approachhas been proposed in [59, 19] and provide an overall accuracy of O(1). Therefore, inour experience, such scheme may work well only in cases where the electromagneticfield is almost constant. We shall now derive better approximations to the adjoint ofthe curl that involve more complex stencils.

3. Discretization of the adjoint curl. The discretization of the adjoint ofthe curl is derived in a very similar manner to the discretization of the grad (whichparallels as the adjoint of the div). The discretization maps faces onto edges and justlike with the grad, difficulties arise for faces who have no neighbors as demonstratedin Figure 3.1.

In order to obtain second order accuracy for the adjoint of the curl in 3D, weintroduce a ghost face 3. Using a ghost face we obtain the standard second order

3As stressed earlier for higher order approximations several ghost faces are needed

8 L. HORESH AND E. HABER

Fig. 2.3. Discretization of edge variables

1u

2u

3u

4u

Fig. 3.1. Example of a hanging edge (marked in bold red)

accuracy stencil of the discrete adjoint of the curl. The overall discretization error isdetermined by the quality of the interpolation used to approximate the ghost points.It is easy to verify, that if a scaled transpose of the curl is employed then for somegrid topologies the interpolation error is O(1). This is where the discretization of thecurl significantly differs from the discretization of the grad. Obtaining either a firstor a second order approximation bring about a loss of symmetry. Since incorporationof second order interpolation is only slightly more complicated than that of the firstorder we turn directly to that option. Six different cases need to be considered; these

2nd ORDER OCTREE DISCRETIZATION 9

cases are illustrated in Figure 3.2. We now discuss each case (Figure 3.2):Case 1: both faces surrounding an edge are of the same size.Case 2: one face is larger than another.Case 3: hanging edge (has only one face) with two equally-sized shifted-adjacent faces.Case 4: hanging edge with two larger shifted-adjacent faces.Case 5: hanging edge with one larger and one equally-sized shifted-adjacent face.Case 6: hanging edge with one equally-sized and one larger shifted-adjacent face.

3.1. Case 1. Case 1 is the standard case where short difference is used on aregular grid and no ghost faces are needed. This configuration yields the standardfinite difference formula

ux =u1 − u2

h+O(h2) (3.1)

3.2. Case 2. In this case, all the faces are located on the same plane, yet, oneface is larger than another. This situation is identical to the discretization of the gradin 2D, and therefore we shall employ the interpolation

uG = u3 +u1 + u2 − 2u3

3+O(h2) (3.2)

3.3. Case 3. In case 3 a hanging edge needs to be treated. However, the ghostface can simply be interpolated by

uG =u2 + u3

2+O(h2) (3.3)

3.4. Case 4. This case is similar to the previous one, only that now the nearestoff-plane faces that are positioned in minimal distance from the edge center on thedifference direction, are the two large faces.

As before, these faces are selected along with the single face that share the edgeand the face which shares the continuation of the hanging edge. This setup entailsthe interpolation formula

uG = u1 +u1 + u2 − u3 − u4

3+O(h2) (3.4)

3.5. Cases 5 and 6. In the last two cases, the off-plane faces are of imbalancedsizes. Nevertheless selecting these face centers along with the additional two faces (onethat shares the edge and the one that shares the continuation of the edge) providessufficient support for the interpolation. By Taylor expansions it is evident that thederivative at the edge center for cases 5 and 6 are essentially attained by identicalexpressions

uG = u1 +3u1 + u2 − 2u3 − 2u4

5+O(h2) (3.5)

4. Solving the resulting linear system.

4.1. The discrete system to be solved. We shall now revert to the discretespace formulation and discuss the solution of the linear system. There are two out-standing issues that require careful attention. First, the loss of symmetry in theresulting system and second and more importantly, the loss of some of the mimeticproperties.

10 L. HORESH AND E. HABER

2u

1u

1u

2u

3ugu

Case 1 Case 2

1u

2u

3u

gu

1u

2u

3u

4u

gu

Case 3 Case 4

1u

2u

3u 4

ugu

1u

2u3

u

4u

gu

Case 5 Case 6

Fig. 3.2. The six cases to be considered for the discretization of the adjoint of the curl

2nd ORDER OCTREE DISCRETIZATION 11

Let C be a discrete form of the curl operator ∇× mapping cell edges to cell faces.Respectively, let C? be the discretization of the adjoint of the curl operator, actingfrom cell faces to cell edges (assuming ghost points are eliminated). Let M , S and qbe discrete form of the magnetic permittivity µ, the dielectric admittivity γ and theexternal current density source ~q. The matrices M and S are obtained by appropriatematerial averaging (see details in [19]). Denoting the discrete forms of the electric andmagnetic fields by e and h respectively, Maxwell’s equations (2.1) in their discreteform can be expressed as

Ce = −iωMh (4.1a)

C?h = Se + q (4.1b)

An equivalent second order system can be obtained by elimination of either e orh. Here, we consider elimination of h and thereby obtain the system

(C?M−1C + iωS)e = −iωq (4.2)

The linear system is similar to the one obtained on staggered regular grid with C?

replacing C>. Since in general, C? 6= C>, the system is not symmetric. However,since the number of local topologies is limited, by inspection, it is possible to show thatthe matrix C?M−1C+εI is an M-matrix for every ε > 0. Unfortunately, this propertyby itself is insufficient for solution of the system. As usual, the main difficulty withMaxwell’s equations is the non-trivial null space of the curl operator. Recall that ifGn

is the nodal gradient matrix then CGn = 0. For low frequencies, the term ωS is smalland the term C?M−1C dominates the linear system yielding a highly ill-conditionedmatrix. Nevertheless, since the null space of the matrix is known, its eliminationis straightforward. Two somewhat similar approaches have been proposed in theliterature to address this regard. In the first, a multi-grid algorithm was developedfor the system, using the null space of the discrete curl to construct an appropriatesmoother [20, 21]. A second approach is to alleviate the null space explicitly by pre-processing the system, then the system can be solved using some iterative method,possibly by multi-grid [16, 2, 23]. Our preferred method is the latter as it offers greaterflexibility in the solution of the problem.

To this end, we use the discrete Helmholtz decomposition

e = −Gnφ− iωaG>na = 0

Here a is the (discrete) vector potential and φ is the (discrete) scalar potential. Thecondition G>na = 0 is the discrete Coulomb gauge condition. Substituting the decom-position into the system (4.2) we obtain(

C?M−1C + iωS SGn

G>n 0

)(aφ

)=

(q0

)(4.3)

This is a saddle-point like system with a highly ill-conditioned (1,1) block. In thefollowing we shall discuss effective methods for the solution of this system.

4.2. Multi-grid preconditioning. The system (4.3) is a saddle-point problemwith a non-symmetric (1,1) block. Multi-level techniques for similar systems withsymmetric (1,1) block and constant coefficients were discussed in [15]. Here, we pro-pose the use of a different technique based on [17, 19] as well as on the work of [2, 23].To its advantage, this approach works well for jumping coefficients.

12 L. HORESH AND E. HABER

Consider first the case where C? = C>. In this case the system can be written as(C>M−1C + iωS SGn

G>n 0

)(aφ

)=

(q0

)In order to obtain a block diagonally-dominant system, we multiply the first blockrow of the system by G>n . Further, relying on the mimetic properties G>nC

> = 0 andG>na = 0 we obtain(

C>M−1C +GnM−1c G>n + iωS SGn

G>nS G>nSGn

)(aφ

)=

(q

G>n q

)(4.4)

The appeal of this system is due to its strongly diagonally-dominant (1,1) and (2,2)blocks, which can be easily used for the development of multi-grid preconditioners andsolvers [2, 6, 19]. In particular, it is possible to use the black-box multi-grid method[10] and alike [29] to effectively solve the system.

The intricacy for our 2nd order formulation ensue since C? 6= C> and therefore,similar manipulation is impossible. Nonetheless, by construction it is obvious that forall entries that correspond to regular interfaces we maintain C>ij = C?

ij , while we are

left with C>ij 6= C?ij for all uneven cell faces.

We therefore utilize the system (4.4) for preconditioning the desired non-symmetricsystem, obtaining

M =

(C>M−1C +GnM

−1c G>n + iωS SGn

G>nS G>nSGn

)−1(I GnM

−1c

G>n 0

)Consequently, the preconditioner requires the solution of the system(

C>M−1C +GnM−1c G>n + iωS SGn

G>nS G>nSGn

)(sψ

)= rhs

at each iteration. This operation is computationally expensive, however, effectivemulti-grid methods have been successfully applied for such systems [19, 2]. We cantherefore replace the inverse of the matrix with a single V or W multi-grid cycle.

4.3. Refinement criteria. We complete the discussion on the solution of thesystem by combining the solver with a multi-level refinement criteria. Many possiblecriteria can be considered for the refinement of the system, e.g.[40, 57, 14]. Here weused the τ criteria (see [56, 1]). Let us denote our linear system (4.3) as Lu = b, anddenote by IHh the restriction operator that restricts from edges of the fine grid h toedges of the coarse grid H.

The τ extrapolation defect criterion quantifies the loss of accuracy due to coars-ening [56]

τHh = LHIHh uh − IHh Lhuh

where the subscripts and superscript h and H denotes relevance to the fine and coarsegrid respectively. In this study, we refine our grids based on this criterion.

5. Numerical studies.

2nd ORDER OCTREE DISCRETIZATION 13

Eanalytic

on 1283 grid

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Enumerical

on 1283 grid

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 5.1. Analytic solution of the electric field E for grid dimensions of 1283 on an OcTreestructure (left) and the numerical solution of the electric field E for grid dimensions of 1283 on anOcTree structure (right), presented as series of cut-planes

5.1. Testing numerical accuracy. In order to evaluate the performance andaccuracies of the proposed discretization and solution scheme, numerical results werecompared with analytical solutions. In the first stage several analytic functions wereset to represent the electric field, in particular we have used

~E =

fx(x, y, z)fy(x, y, z)fz(x, y, z)

in the interval [0, 1]3 where

f = exp

(− (x− x0)2

a− (y − y0)2

b− (z − z0)2

c

)where a = b = c = 0.01, x0 = 0.01, y0 = 0.02, z0 = 0.03 and assumed that thecoefficients in Maxwell’s equations were σ = µ = ω = 1.

An OcTree structure was generated based upon the analytic right hand side ofthe aforementioned Maxwell’s equations. Further, differential operators were con-structed for that OcTree structure and the resulted linear system was solved. Wethen compared the numerical solution to the analytic one. The analytical and numer-ical solutions for grid size of 1283 and 61440 OcTree grid cells are presented in figures5.1. We took care to ensure that representation of all the aforementioned topologieswere considered. For an instance, the enumeration of x-edges w.r.t. y faces per classwere: 59298, 4148, 36, 1984 and 1784 jointly for cases 5 and 6.

As depicted by the two images, at least qualitatively the numerical solution bearsgreat resemblance to its analytical counterpart.

Encouraged by the above results, a more quantitative study was conducted. Thisvalidation study was set to confirm that the prescribed asymptotic accuracies wereobtained in effect numerically. For that purpose, the truncation and solution errorof an OcTree structure of grid size 163 and 243 cells were computed. Next, the gridwas uniformly refined (i.e each cell was divided into 8 smaller cells), and the processrepeated up to a grid size of 2563. In Table 5.1 the truncation errors as well asthe solution errors are displayed. As expected, consistent truncation errors of O(h)and solution errors of O(h2) were exhibited respectively. These results confirm thetheoretical accuracy predicted in Section 2.

14 L. HORESH AND E. HABER

Grid size Smallest cell Largest cell Truncation error Solution error163 1 4 4.20 6.7323 2 8 2.19 4.16643 4 16 1.11 1.251283 8 32 0.557 0.312563 16 64 0.279 0.08

Table 5.1Convergence of the truncation error and solution error for the test problem. Note that trunca-

tion error is O(h) while solution error is O(h2).

Grid size Unknowns ω = 102 ω = 103 ω = 105

[Iterations] [Iterations] [Iterations]163 120 10 10 6323 960 15 15 7643 7680 21 21 61283 61440 26 29 62563 491520 20 29 8

Table 5.2The number of preconditioned GMRES iterations for a range of grid sizes and frequencies.

5.2. Results for preconditioning. Once the results were qualified quantita-tively, the next stage would be to ensure that solutions can be obtained reasonablyfast. Here again we used the analytical solutions over OcTree grids. We then gener-ated analytical right hand sides, for which we solved the systems. We examined thepreconditioning scheme by recording the number of Krylov-subspace iterations neededfor solution of a prescribed accuracy (relative residual error of 10−6) over problems ofdifferent dimensions. We have repeated the study for a range of frequencies, to allowfurther assessment as for the robustness of the scheme. The results are summarizedin Table 5.2.

The number of iterations required for solution of the system is almost mesh sizeindependent, as expected theoretically when an approximation of similar eigenvaluesspectrum is used as a preconditioner.

As the frequency increases the significance of the stabilizing term iωS of the (1,1)block becomes more profound, and therefore dictates lower (yet, consistent) numberof iterations for the solution of the system. Similar numerical trait was observed formixed finite elements discretization [47].

5.3. Solution in the presence of discontinuities. Next, we observe the be-havior and stability of the solution as discontinuities emerge. For that purpose, wegenerated OcTrees of increasing dimensions based on a 3D MR phantom [46] (seeFigure 5.2). We then assigned conductivity values according to the gray-scale valuesof the phantom, and rescaled their values by logarithmic factors ranging from 101 to104, to simulate growing level of coefficients’ discontinuity. All systems were solvedto a relative residual error tolerance of 10−6. This particular test problem is ratherdifficult as the skull (represented as a white ring in Figure 5.2) is a thin structureof high contrast. Resolving such fine structures adequately requires very small cells.The results for this numerical study are summarized in Table 5.3.

While for discontinuities in the order of 10, mesh-independent convergence is stillretained, for larger contrasts of 100 and 1000, the solution was no longer grid inde-

2nd ORDER OCTREE DISCRETIZATION 15

Fig. 5.2. Transverse 2D slices of the 3D phantom model

Grid size Unknowns Discont. ×10 Discont. ×102 Discont. ×103 Discont. ×104

[Iterations] [Iterations] [Iterations] [Iterations]163 960 9 27 36 37323 4656 8 45 46 46643 20840 10 87 85 861283 82496 9 48 97 37

Table 5.3Number of GMRES iterations required to solve the system, for a range of grid sizes and degrees

of discontinuity.

pendent, albeit, the increment in iteration count still exhibits somewhat conservativegrowth. For extremely large discontinuities optimal convergence can be obtained bydeveloping an appropriate induced prolongation and restriction operators [56].

The above studies were conducted on a quad-core, 8GB RAM workstation usingMatlab 2010a under Windows 7 64 bit operating system.

6. Conclusions and future challenges. In this study we have developed anO(h2) discretization scheme for Maxwell’s equations on OcTree grid structures. Themulti-grid preconditioning scheme proposed here displayed mesh-independent conver-gence behavior over a range of frequencies as well as for mild discontinuities.

The proposed strategy for construction of differential operators on OcTree struc-tures can be readily applied for a multitude of PDEs problems, where obviously careand consideration must be dedicated to the individual structure and linear algebra ofthe problems under hand.

In this study, OcTree structure was the topology of choice for generation of or-

16 L. HORESH AND E. HABER

thogonal adaptively refined grids. Notwithstanding, other consistent splitting choices,rather than binary per dimension as in [64, 8] may be sought. Such choices may pos-sibly lead to more favorable discretizations or numerical schemes for specific domainproblem. Since the proposed discretization scheme relies on construction of standardfinite difference stencils (which can be defined independently from the physical grid),as well as on selection of suitable interpolation support for the ghost points, the pro-posed approach can be applied for alternative partitioning schemes straightforwardly.

The discretization scheme presented in this study in its generalized form shown in2.6, can be utilized for higher orders, provided that suitable difference template sup-port (including ghost points) as well as appropriate interpolation condition accuraciesare satisfied.

Given that only local topologies are analyzed for the construction of the differ-ential operators, matrix-free processing is indeed feasible. Nevertheless, since theresulted operators are sparse, and exploitation of their structure is of tantamountimportance, the motivation in pursuing such research avenues in the context of PDEoptimization is not entirely clear yet.

There are still several open questions which can be considered for future endeavorsin the field: firstly, from a critical perspective, one may question whether anO(h2) dis-cretization error yields adequate level of accuracy in practical settings. Obviously, forsmooth regions the answer is indeed positive, however, our numerical studies indicatethat higher levels of accuracy may be desired when truncation errors are significant.Secondly, one may wish to consider alternative formulations that may either simplifyimplementation, grant better accuracies, or be more favorable for current solutionmethods for large-scale problems.

7. Acknowledgements. The authors wish to express their great gratitude toAndrew Conn and Raya Horesh for their detailed remarks and thorough reviews. Inaddition, we wish to thank Michele Benzi and Ulisses Mello for their valuable advices.

This research was supported by IBM TJ Watson Research Center as well as byNSF grants DMS-0724759, CCF-0427094, CCF-0728877 and by DOE grant DE-FG02-05ER25696 and by NSERC IRC grant.

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