multiscale cell-based coarsening for discontinuous problems

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Mathematics and Computers in Simulation 79 (2009) 1915–1925

Multiscale cell-based coarsening for discontinuous problems

Alfonso Limon a, Hedley Morris b,∗a School of Mathematical Sciences, Claremont Graduate University, 710 N. College Ave., Claremont, CA 91711, United States

b Department of Mathematics, San Jose State University, One Washington Square, San Jose, CA 95192, United States

Received 31 August 2006; received in revised form 19 June 2007; accepted 21 June 2007Available online 24 July 2007

Abstract

Whether tracking the eye of a storm, the leading edge of a wildfire, or the front of a chemical reaction, one finds that significantchange occurs at the thin edge of an advancing line. The tracking of such change-fronts comes in myriad forms with a wide variety ofapplications expressible as PDEs. Expanding on Ami Harten’s ideas, we construct an alternative to wavelet-based grid refinement,a multiresolution coarsening method that is capable of capturing sharp gradients across different scales, thus improving PDE-basedsimulations by concentrating computational resources where the solution varies sharply. We present this alternative grid coarseningmethod and compare its performance to other multiresolution methods by means of several examples.© 2007 Published by Elsevier B.V. on behalf of IMACS.

PACS: 42C40; 65M50

Keywords: Generalized wavelets; Grid refinement; Multiresolution analysis

1. Introduction

There are many adaptive wavelet-based PDE methods [3,8–10,19,21,25]. Our alternative method, an expansion ofAmi Harten’s generalized wavelets [18], yields a multiresolution coarsening procedure that captures sharp gradientsacross different scales and improves PDE-based simulations by concentrating computational nodes where the solutionchanges abruptly. Our method has also proved useful in flagging points near jumps that would benefit from adaptivestencil selection strategies, such as those proposed by Harten and Osher [20,23]. In the present paper, we use a linearversion of Harten’s multiresolution analysis [1,30] to construct a multilevel based front tracking scheme. This scheme,dubbed the Multilevel Front Tracking method, or MFT, works via detail coefficient thresholding [15]. These detailcoefficients are used to adapt the grid near the jump condition, thereby providing a coarser version that capturesthe essential features of the original solution. Our MFT grid coarsening scheme thus provides accurate derivativeinformation to solve PDEs with discontinuous solutions. The goal of the present project is to provide an adaptivecomputational platform based on generalized multiresolution analysis that is able to discretize a PDE, refine thesolution and solve the resulting linear system all under one unified framework.

Section 2 introduces Harten’s multiresolution analysis and proceeds to a detailed description of our MFT method’sdiscontinuity localization strategy in Section 3. Sections 4 and 5, respectively, cover the localization of jumps after the

∗ Corresponding author.E-mail addresses: [email protected] (A. Limon), [email protected] (H. Morris).

0378-4754/$32.00 © 2007 Published by Elsevier B.V. on behalf of IMACS.doi:10.1016/j.matcom.2007.06.011

mailto:[email protected]

mailto:[email protected]

dx.doi.org/10.1016/j.matcom.2007.06.011

1916 A. Limon, H. Morris / Mathematics and Computers in Simulation 79 (2009) 1915–1925

tiling of the computational domain and the development of the multilevel front tracking method. Finally, in Section6, our MFT method is compared with other multilevel methods both as a coarsening strategy and later as an adaptivePDE solver. We end with a short summary and discussion of our development road map in Section 7.

2. Multiresolution analysis

Multiresolution schemes are a natural framework by which to locate fast transitions across multiple scales. Wavelets[11,15,24,31] and their Lifted extensions [13,22,32,33] form the basis for many multiresolution schemes. In additionto these, there are nonlinear counterparts to Lifting [6,7,16,17,26] and adaptive stencil selection methods, such as thosefirst proposed by Harten [18] and later extended by Arandiga and Donat [1] and Schroder-Pander et. al. [30]. We usethese extensions to construct a cell-based coarsening method, and what follows is a short introduction to Harten’smultiresolution analysis.

Suppose that we have a space of functionsF ⊂ {f |f : Ω ⊂ Rm → R}, where Ω is a bounded region, and that ∀f ∈Fthere is a discretization operator Dk : F → Vk, where Vk is a finite linear space with dimension k. The objective isto design a multiresolution scheme specifically adapted to sequences obtained from Dk. This is achieved through theintroduction of reconstruction operatorsRk : Vk → F. The operators,D andR, can be constructed to provide differentresolution details

(2.1)

where dim(Vk−1) < dim(Vk).Given Vk and Vk−1, a decimation operator is defined Dk−1

k : Vk → Vk−1, which lowers the resolution level fromk to k − 1. Inverting the process, a prediction operator Pk

k−1 : Vk−1 → Vk increases the resolution from k − 1 to k.Therefore, the multiresolution framework

(2.2)

acts on a sequence vk ∈ Vk constructed by a discretization process vk = Dkf at resolution level k, and by definition,Dk−1

k (Dkf ) = Dk−1 f , so Dk−1k = Dk−1 Rk. However, the sequence is nested, Dkf = 0 ⇒ Dk−1f = 0, ∀f ∈F, so

Dk−1k cannot depend on the reconstruction operator Rk; thus, Dk−1

k is linear. Furthermore, the prediction operator Pkk−1

is the right-inverse of Dk−1k and so Pk

k−1 = DkRk−1.

Using this framework, vk ∈ Vk can be approximated via the information content at level k − 1, i.e., Pkk−1D

k−1k :

Vk → Vk, and errors in the approximation can be computed by ek = (IVk − Pkk−1D

k−1k )vk. This constructs a one-

to-one correspondence between vk and [ek, vk−1], but ek is in the null space of Dk−1k , because Dk−1

k ek = Dk−1k vk −

(Dk−1k Pk

k−1)vk−1 = 0. As a consequence, expressing ek in terms of a basis in Vk results in redundant information.

However, this redundant information can be discarded by projecting onto N(Dk−1k ) and expressing the prediction

error as ek = ∑jd

kj μk

j ≡ Ekdk, where μk

j spans N(Dk−1k ). Defining the assignment operator Gk as EkGk = IN(Dk−1

k)

results in a less redundant correspondence between vk and [dk, vk−1], where the detail coefficients dk = Gkek arethe projection errors expressed by any basis of N(Dk−1

k ). In other words, dk represents the information at level k thatcannot be predicted by Pk

k−1 at level k − 1. The pyramid scheme,

(2.3)

expands this two-level scheme to multiple resolution levels; refer to [1,30] for further details. In Section 4, we developa coarsening strategy for mesh refinement based on dk as a variational measure of f at scale k, but first, the jump-pointsmust be localized.

A. Limon, H. Morris / Mathematics and Computers in Simulation 79 (2009) 1915–1925 1917

3. Locating discontinuities

Utilizing the techniques in Section 2, we motivate cell-based coarsening by constructing first a jump localizationstrategy on dyadic grids in one dimension. This jump location strategy is then extended into Rn in Section 3.2, andculminates in the cell-based coarsening strategy detailed in Section 4.

3.1. Locating jumps in R

We restrict Harten’s multiresolution framework to a compact dyadic sampling of R and suppose the discontinuitieshave codimension-one and exist on non-dyadic points. This construction assures that any jump in f has two neighboringpoints on the dyadic mesh Vk and that one of the points, when projected to Vk−1, is in the detail branch of themultiresolution transform. Moreover, this point in the detail branch can be located by following Donoho’s thresholdingstrategy [15]. The second neighboring point to the discontinuity can be located by applying the same procedure to thescale branch of the transform, as occurs in wavelet packets [36]. Consequently, we are able to locate both neighbors inVk that enclose the discontinuity. What follows is a more detailed account of this process.

As demonstrated, Harten’s multiresolution framework is capable of down-sampling and removing redundancies.Hence, this procedure is akin to grid coarsening, where some redundant portions of the solution are discarded tominimize computational cost. In most applications, the grid will be updated countless times in a single simulation,so the update procedure must be computationally inexpensive. Therefore, the down-sampling is accomplished usinga Lazy Wavelet, so that Dk−1

k becomes an injection operator [4,35]. The projector, Pkk−1, was chosen to be linear, as

this guarantees that the interpolation coefficients are never larger than one and thus remain stable. The correspondingassignment operator Gk = [

−→0 |Dk−1

k |−→0 ], where−→0 is a column vector of all zeros, and Ek is simply the transpose of

Gk [30].In keeping with this computationally minimalist mantra, note that the action of both Dk−1

k and Gk on vk does notrequire any computations, as the operators simply discard even or odd entries, respectively. Hence, the detail coefficientsare computed as

dk = Gk(I − Pkk−1D

k−1k )vk (3.1)

and the refinement measure on a dyadic grid is 2k|dk|. Hence, given a discontinuous f on Vk, compute transform (3.1),and whenever 2k|dk| > tol label vk

j a discontinuity-neighbor, where tol is a predefined approximation tolerance.One way to locate the second neighboring point of the discontinuity would be to apply the same procedure to the

scale coefficients of Vk, thus computing a full wavelet packet, but this is not necessary. By design Pkk−1 has a compact

stencil, so as long as both discontinuity-neighbors in Vk are more than two grid points apart from each other, the secondneighbor can be uniquely identified. One need only then locate vk

j a discontinuity-neighbor, switch from the detail to

the scale-branch and compute |dkj−1/2| and |dk

j+1/2|; whichever magnitude is larger defines the second neighboringpoint to the discontinuity.

Algorithm 1. Locating jump boundaries in R

This localized search for the second discontinuity-neighbor in Algorithm 1 lowers the complexity from O(2N)to O(N + J), where N is the number of data points and J the number of jumps in Vk. Furthermore, as N → ∞,J/N → 0, so the extra work to localize the second neighboring point becomes negligible for large problems. Thisscheme achieves the same complexity as Chan’s ENO-Wavelet method [5] and is similar in spirit to another of Harten’smethods [19], in that we too use a hierarchy of nested grids obtained by diadic coarsening to compute an equivalentmultiresolution representation to find jump-discontinuities. In the following section, we expand on this one-dimensionaldiadic coarsening strategy to Rn.


Fig. 1. Cells Ck (�), dual-cells Ck(�), discontinuity (�).

3.2. Locating jumps in Rn

Following the construction in Section 3.1, the discontinuities have codimension-one, Vk is dyadic in each coordinate,and jumps occur on non-dyadic points. Recall that Algorithm 1 returns two points in Vk that bound the jump-pointin R. However, in Rn for all n > 1, the location of the detail coefficients are, in general, no longer collinear to thediscontinuities, and thus the correspondence between the norm of the detail coefficients and the jump locations is nolonger one-to-one. This problem can be circumvented by using n-orthogonal one-dimensional transforms to locatethe discontinuity-neighbors in Vk ⊂ Rn by taking advantage of the coordinate direction that produced each detailcoefficient.

Given Vk, we construct a tiling in terms of cells Ck. Each cell (or voxel) contains 2n vertices; assigned to each isan element vk. Each edge in Ck is assigned a detail coefficient; however, dk /∈ Vk, so a one-to-one correspondencebetween edges and details is not possible. But by using n-orthogonal one-dimensional transforms, we are able toachieve a correspondence between edges and details. In addition, the n-transforms encode the projection error alongthe standard basis, {e1, . . . , en} ∈Rn, serving as an error flux across cell edges.

Suppose f is a discontinuous surface on Vk ⊂ R2 and f (:)|e1 ⊂ R is a multi-vector representation of f sorted

according to direction e1. Applying Algorithm 1 to f (:)|e1 labels all cell edges parallel to e1 whose |dk,je1 | > tol/2k.

Repeating the same procedure along e2 completes the labeling of all edges that intersect discontinuities. Note that hadwe retained all the detail coefficients dk

e1and dk

e2, each cell edge in Vk would be assigned a detail coefficient depending

on the orientation of the cell edges, and this scheme would mimic Harten’s multiresolution analysis, as presented inSection 2. However, as also noted by Muller [25], we need not store all detail coefficients to determine if a cell containsa discontinuity.

Note that any codimension-one discontinuity passing through Ck ⊂ Rn must cross at least one edge, and thereforeCk will contain a |dk

ei| > tol/2k. Hence only one d

k,jei ∈ Ck is required to determine if the cell contains a discontinuity.

This simplifies our coarsening strategy by eliminating the need to store all dk,jei , and thus the labeling of each node as

discontinuous can be performed by labeling a single node on a dual-grid [29]. In other words, whenever 2k|dkei| > tol,

the dual-cell Ckis marked. Fig. 1 illustrates this grid/dual-grid construct in R2.

In general, given a discontinuous surface f sampled on Vk ⊂ Rn, we can localize all edges in Ck intersected by

a codimension-one discontinuity, and any one of those edges can be used to assign Ckas discontinuous. This is

accomplished by applying Algorithm 1 to array f (:)|ein-times, and thereby encoding all discontinuities in Vk onto the

dual-grid⋃

τ Ckτ , where τ enumerates each cell Ck that tiles Vk. The identification of cells containing discontinuities is

expressed in the following algorithm.

Algorithm 2. Locating jump in cells


The next task is to discard as many elements as possible in Vk that are in smooth regions, i.e., elements that do not

neighbor nodes in Ck. In Section 4, we explore a stable coarsening procedure for Vk that maximizes grid sparsity.

4. Cell-based coarsening

In Section 3, we presented a method to locate jumps and label cells that contain discontinuities. In this section, we

coarsen Vk using the information in each Ckand add a constraint on the largest size a neighboring cell can have based

on homogenization and numerical experiments. The resulting grid has a balanced-quadtree structure, which boundsthe growth in the wavenumbers across neighboring cells.

Let δkj = xk

j+1 − xkj represent the distance between vj+1 and vj at level k and define γk = maxj(δk

j+1, δkj−1)/δk

j as ahomogenization measure of a multilevel grid at level k. In one dimension, Ingrid et al. showed that bounding γ ≤ 2.4992controls the approximation properties of a third-order subdivision scheme (refer to Lemma 8 and Appendix D in [12]).Moreover, Berger and Colella have shown numerically that when γ = 2, the resulting AMR grids provide high-qualityresults in two and three dimensions [2]. Using the more conservative integer estimate for γ , we constrain the sizeincrease (or decrease) of any neighboring cell.

Constructing the sparse grid S ⊂ Vk is achieved by reinterpreting our localization transform in R2 as a waveletpacket, so that all subbands form a quadtree and individual subbands are in one-to-one correspondence with rectangular

regions in the wavenumber space [36]. This implies that Ckcan seed the formation of an MX-Quadtree [28], whose

depth D ≤ log(L/δk) + 3/2, where δk ∈ Ck has an active node in Ckand L is the side-length of Vk

n×n. Performance-wise the quadtree is also advantageous, as a quadtree of depth D storing N points can be constructed and balanced inO((D + 1)N) time [14], where the balanced constraint on the tree enforces γ .

Although we motivated the quadtree construction in R2, the tree structure can be extended to three dimensions,becoming an octree, or more generally, a Kd-tree in K-dimensions [27]. Therefore, given a Vk ⊂ Rn, we can alwaysgenerate a sparse grid S ⊂ Vk via a balanced Kd-tree. Algorithm 3 sketches such a procedure:

Algorithm 3. Constructing a Sparse Vk

The next challenge is to compute derivatives on S ⊂ Vk and evolve the solution in time, which include the possibleinterpolation of values from S back to Vk. These issues are resolved in the next section and yield our Multilevel FrontTracking Method.

5. Multilevel front tracking

Following the construction of the sparse grid S in Section 4, derivative information for each node in S must becomputed in order to use our coarsening method to solve PDEs. Perhaps the simplest method to compute derivativesbased on local function values is to use Finite Differences [34]. However, this method assumes that the discrete functionvalues are sampled from a continuous function, so care must be taken to avoid the application of the differentiationstencil across jumps. In places where a jump is present, we will use an adaptive stencil selection criterion based on avariant of the ENO scheme developed by Harten et al. [20].

As a consequence of the multiresolution analysis of Section 2, each cell Ck ∈S has a smoothness measure associatedwith it that depends on the largest δk ∈ Ck. Moreover, by the construction of S in Section 4, any discontinuity detectedwill belong to the set of unit-cells, i.e., the set of smallest cells in the quadtree. Therefore, all Ck that are larger thanthe unit-cell size are by definition not discontinuous, so we can apply Finite Differences to compute the derivatives.

In places where the solution changes abruptly due to a discontinuity or boundary layer, the cell Ck will contain an

associated node in Ckand have unit-cell size. For those discontinuous cells, we apply an essentially non-oscillatory

scheme to compute derivatives [23]. Furthermore, using the same argument as in Section 3.1, we can show that for


Fig. 2. Coarsening f—(a) via wavelets; (b) sketch of f; (c) via MFT.

large enough domains, the extra computational cost required to compute derivatives accurately in discontinuous cellsis negligible. Thus the overall cost of computing derivatives depends mainly on the computational cost associated withthe Finite Difference derivatives on S.

Following the same logic, the interpolation stencil should not cross discontinuous cells. But due to Algorithms 2and 3, all nodes neighboring discontinuities are not discarded when Vk is coarsened, so no interpolation is required.This implies that any node requiring interpolation is in a smooth region and thus free of the Gibbs effect. To illustratethis point, Fig. 2 contrasts classical wavelet coarsening in Fig. 2(a) with our MFT coarsening in Fig. 2(c). To aid ourintuition, a sketch of the discontinuous function f and its associated vk is given in Fig. 2(b).

The main illustrative point behind the sketch is four-fold:

[1] Classical wavelets coarsen according to a predefined Lazy (or Haar) wavelet, thus resulting in a coarsening strategythat is blind to abrupt changes in f.

[2] MFT using only linear operators is able to account for abrupt changes in f when coarsening Vk.[3] Classical wavelets interpolate crosses jumps in Vk causing the Gibbs effect.[4] MFT does not interpolate across jumps, so no Gibbs effect occurs.

Therefore, MFT is an adaptive linear scheme with many of the same features that other nonlinear multiresolutionschemes possess [6,7,16,17,26], but which does not incur the extra computational cost associated with using nonlinearmethods. In the next Section 6, we apply MFT to several piecewise-continuous functions, and with a slight addition,we can use MFT to solve Sod’s shock-tube problem and Burgers’ equation.

6. Numerical experiments

Building upon the MFT method introduced in Section 5, we explore the approximation capabilities of MFT inSection 6.1 and compare it to other multiresolution methods in the literature. In Section 6.2, the MFT method isextended to allow the method to handle numerical solution of PDEs with discontinuities.

6.1. Approximating piecewise smooth functions

We begin by comparing the approximation capabilities of MFT to other multiresolution methods. A function, f,proposed by Chan and Zhou [5] is illustrated in Fig. 3 (top), a coarser version achieved via MFT, f , is shown in themiddle subfigure, and the absolute error between both is plotted in the bottom subfigure.


Fig. 3. Chan’s f —(top) original; (middle) approx. fk=5; (bottom) abs. error.

The original function f ⊂ Vk is uniformly sampled using 1024 points. The application of Algorithm 1 to f returnsall the discontinuity-neighbors satisfying |dk| > 0.02, which are then used by Algorithm 3 to construct a sparse gridS. Restricting the tree to five levels results in f ⊂ S an approximation to f requiring only 170 points. Projecting fromS to Vk, MFT achieves ‖f − f‖∞ ≈ 4 × 10−14 on Chan’s function.

Taking the 170 points as a baseline for comparison, we repeat the approximation ‖f − f‖∞ using various mul-tiresolution methods and list the results in Table 1. Using classical wavelets (i.e., Haar, Daubechies D4 and D6) resultsin large approximation errors due to the Gibbs effect near the jumps (first column). These errors can be reduced byincorporating ENO-type function extensions to the scaling (or wavelet) coefficients near the jumps in Vk, as proposedby Chan and Zhou [5]. These extensions reduce the approximation errors according to the accuracy of the methodused to extend Vk across the jumps. This phenomenon can be seen in the second column of Table 1, where Haar-ENOis essentially first order, while DB4-ENO is fourth order and DB6-ENO is sixth order accurate. The third columndepicts the approximations achieved by Harten’s methods: the first ENO-SR method is based on point values, whilethe second is based on cell-averaged values. Both use single transform methods and do very well, but MFT, which usestwo transforms, out-performs both.

The same function approximation procedure can be used to coarsen piecewise smooth surfaces. A sample surfaceis illustrated in Fig. 4. In two dimensions, we apply Algorithm 2 to find all cells in Vk that have a discontinuity, andthen apply Algorithm 3 to return the appropriate coarse version f ⊂ S. A key feature of MFT can be illustrated easilyin R2 by plotting all the detail coefficients [dk

e1, dk

e2] and comparing them to the quadtree grid (refer to Fig. 5(a and b),

respectively). Note that while [dke1

, dke2

] does not produce a good tiling of Vk, due to the non-uniform distribution ofdetail coefficients, MFT does.

In this section, we used MFT to reconstruct functions in one and two dimensions and showed that S tiles Vk well.In the next section, we build upon the good tiling properties of S and present some preliminary results associated withhyperbolic PDEs.

Table 1Comparing MFT to classical wavelets, ENO wavelets and Harten’s methods

Traditional Wavelet-ENO Harten type

Haar 1.E+00 Haar-ENO 8.E−02 ENO-SR point 1.E−07DB4 1.E+00 DB4-ENO 1.E−04 ENO-SR cell 1.E−09DB6 1.E+00 DB6-ENO 2.E−06 MFT 4.E−14


Fig. 4. Fine-scale surface f ∈ Vk .

6.2. Multilevel solver

In Section 6.1 we showed, by means of numerical experiments, that MFT can be used to coarsen both curves andsurfaces. We now build on these ideas and develop a grid coarsening algorithm that can be used to solve numericallyPDEs exhibiting discontinuous solutions. In our first experiment, we solve Burgers’ Equation using MFT as a gridcoarsening strategy and select between FD and WENO derivatives depending on the local smoothness of the solution,as detailed in Section 5.

At time t0 two shocks begin traveling to the right and eventually merge. Fig. 6 shows a snapshot of the solution attime t = 0.079. The top plot serves as a reference solution having 1024 uniform points and all derivatives are computedusing WENO. The middle plot displays the solution of Burgers’ equation using MFT; the points selected for WENOdifferentiation are marked by circles. Both solutions are in good agreement and reach the merging point at the sametime. The bottom plot displays the compression history between the reference solution and our MFT method.

In the second experiment, we use MFT to provide a new grid at each step of the solution of the Sod’s shock-tubeproblem. As is evident from Fig. 7, the MFT method produces coarse grid representations of the pressure, density andvelocity that conform well to the solution throughout most of the domain, including regions where the functions arediscontinuous. However, when there is a discontinuity in the derivatives, oscillations in the solution appear (refer tozoom-in region). This is due to the fact that dk senses changes in the function values but not in the derivatives; thusboth the number of points and differentiation method are not optimally chosen in those regions. A simple fix is toincorporate a multi-wavelet framework, but this could hinder the compression severely, so we are currently workingon an alternative solution.

Fig. 5. Comparing how detail coefficients and MFT tile Vk . (a) Detail coefficients via tensor product. (b) 3-Level quadtree grid via MFT.


Fig. 6. Burgers’—(top) WENO; (middle) MFT; (bottom) compression.

Fig. 7. Shock-tube solution—(a) pressure p, (b) density ρ and (c) velocity u.

As the previous numerical experiment shows, the MFT method can be successfully used as both a grid coarseningmethodology and as a selection criterion for nonlinear differentiation methods. This flexibility in the method opensthe door to further development of MFT as an adaptive PDE solver. The one remaining detail is the problem of shockformation: currently, in order to resolve shock, we limit the time step so that the solution never lands on a cell solarge that it is blind to shock development. We think the best alternative solution would be to incorporate the timestepping mechanism into the multiresolution analysis. In this way, the grid coarsening, the differentiation methodand the integration procedure are all built from a common multiresolution “atom,” a technique that would unify theapproach.

7. Conclusions

We have detailed Harten’s generalized multiresolution analysis, and used a linear version of this method to constructa multilevel based front tracking scheme. This scheme, dubbed MFT, is capable of capturing discontinuities in the


solution via detail coefficient thresholding. In addition, the detail coefficients are used to adapt the grid near the jumpcondition, thereby providing a coarser version that captures the essential features of the original solution. Furthermore,the method has been used to flag points near the jump condition that would benefit from further nonlinear analysis.In this way, the grid coarsening method provides accurate derivative information, thus extending the scheme to solvehyperbolic-type PDEs.

The MFT method is general enough to provide a platform for further advancements in the numerical solutionof multiscale PDEs. Expanding on the coarsening technique discussed in this manuscript, we have been developinga discretization procedure that couples with the MFT method, thereby unifying the entire adaptive process into asingle multiresolution framework. Our goal is to provide an adaptive computational platform based on generalizedmultisolution analysis that is able to discretize a PDE, refine the solution and solve the resulting linear systems allunder one unified framework.

Acknowledgements

We thank the referees for their insight, comments and many suggestions, which streamlined and enhanced our finalmanuscript greatly. In addition, we thank Claremont Graduate University for providing the CGU Dissertation Grant tosupport Alfonso Limon throughout this project.

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multiscale cell-based coarsening for discontinuous problems

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