Discrete Physics using Metrized Chains

A. DiCarloUniversità Roma Tre

adicarlo@mac.com

F. MilicchioUniversità Roma Tre

milicchio@dia.uniroma3.it

A. PaoluzziUniversità Roma Tre

paoluzzi@dia.uniroma3.itV. Shapiro

University of Wisconsinvshapiro@engr.wisc.edu

ABSTRACTOver the last fifty years, there have been numerous efforts todevelop from first principles a comprehensive discrete formu-lation of geometric physics, including Whitney’s geometricintegration theory, Tonti’s work on unification of physicaltheories, research on mimetic discretization methods, dis-crete exterior calculus, and Harrison’s theory of chainletsamong others. All these approaches strive to separate phys-ical models into standard topological, geometric, and physi-cal components. While each of these components appear tobe well understood, the effective computational connectionbetween these three components is still lacking, leading todifficulties in combining, reconciling, and refining physicalsimulations. This paper proposes such a connection usingmetrized chains defined on a cell complex, an abstraction ofa decomposition of a Riemannian manifold considering onlytopological-related properties, to establish a discrete metricstructure on top of a discrete measure-theoretical structure,embodied in the underlying notion of measured (real-valued)chains.

The metric structure of the ambient space—be it Euclideanor Riemannian—carries basic information on the physicalphenomena taking place on that scene. Therefore, it is vitalthat this structure be properly mimicked by discrete geomet-ric models meant to be used for trustworthy physics-basedsimulations. The underlying cell complex endowed with a(discrete) measure-theoretical structure may be used to re-produce the measure-theoretical properties of the concretemesh; refining or coarsening a mesh changes both topologyand measure. Next, we establish a direct link between thediscrete structure described by a chain complex and the Rie-mannian metric of the underlying manifold. To this end, weassociate a local p-vector field with each elementary p-chain,and identify the inner product between two chains with theinner product between the corresponding multi-vectors.

By doing so, chains are identified with elements of their dual

space, i.e., cochains, in a way that mimics the metric of theapproximated manifold. Moreover, boundary and cobound-ary operators—acting respectively on chains and cochains—may then be composed with each other, giving rise to physi-cally meaningful Laplace-de Rham operators, an ubiquitousingredient of physical modeling.

Categories and Subject DescriptorsI.3.5 [Computational Geometry and Object Model-ing]: Physically based modeling; J.2 [Physical Sciencesand Engineering]: Engineering

General TermsDiscrete calculus, Geometrical and physical modeling.

1. INTRODUCTIONThe notion of a (finite) cell complex—which we take herefor granted—abstracts the topological features of any rea-sonable computational mesh, stripping away all its extra-topological properties. A cell complex is thus an equiva-lence class of (geometrically different) meshes sharing thesame topology, i.e., the same sets of cells of each dimensionand the same incidence relations. By design, much of theinformation included in a concrete mesh is wiped out in thecorresponding cell complex. The rest of the game consists inrebuilding layer after layer more-than-topological structuressuited to the established discrete setting, i.e., compatiblewith the underlying cell complex.

The first move is to change cells into chains, by attachinga value to each cell. Standard books on algebraic topol-ogy [14,21] pick these values out of any commutative group,since this is the minimum apparatus enabling chain addi-tion. This seems to be done for merely instrumental rea-sons: standard books talk of ‘formal sums’. Our attitude,embryonically showed in [12], is different: we need a fieldof scalars, in order to be able not only to add, but also toscale chains. So, to us a chain complex is the vector spacespanned via linear combination by unit chains, i.e., chainsattaching the unit value to a single cell and the null value toall the others. Thanks to the linear structure imparted tothe set of chains, cochains naturally appear as linear formson chains, i.e., as elements of the algebraic dual of the spaceof chains. In this context, the boundary is a linear operatorand the coboundary is its dual.

Moreover, we wish to impart size to cells. We are thus ledto consider real-valued chains, attaching a signed p-measure

to each p-cell. The underlying cell complex is so endowedwith a (discrete) measure-theoretical structure that may beused to reproduce the measure-theoretical properties of theconcrete mesh topologically represented by the cell complex:by definition, a homeomorphic distortion of the mesh doesnot affect the corresponding cell complex; however, it doesaffect sizes, in general. Refining or coarsening the meshchanges both topology and measure. In particular, if a cellis split into two, a desideratum is that the sum of the sizes ofthe two daughter cells equals the size of the parent cell (or,if the splitting refines the geometry, that the daughter cellsprovide a better approximation to the size of the mimickedpatch than the parent cell). Cochains represent densitieswith respect to the measures imparted to cells by real-valuedchains, and the duality pairing between chains and cochainsrepresents integration in a discrete setting.

Measure, however, does not exhaust geometry. To rein-troduce metric properties, we endow the linear space ofchains of a (proper) Euclidean inner product, i.e., a real-valued nondegenerate (symmetric and positive definite) bi-linear form, canonically identifiable with a bijective linearmap of that space into its dual. Introducing an inner producton chains is tantamount to identifying chains with cochains.Via this identification—which depends on the inner prod-uct selected—boundary and coboundary operators (actingrespectively on chains and cochains) may be composed witheach other, giving rise to the Laplace-de Rham operators.Changing the inner product changes neither the boundarynor the coboundary operators; however, it does change theLaplace-de Rham operators.

We aim to establish a link between this discrete structureand the Riemannian metric of the underlying manifoldM, alink strong enough that the Riemannian metric may be re-covered in the limit of infinite mesh refinement. The keyidea, in some ways dual to the construction of Whitneyforms for simplicial cochains [7,35], is to associate a p-vectorfield on M with each unit p-chain, for p ranging from 0 tod := dimM. This construction is by no means unique. Wewant chains to be smeared into fields so as to satisfy the fol-lowing conditions: i) representativeness: the p-vector fieldassociated with a unit p-chain should be tangent to the cor-responding mesh element; ii) locality: the support of thep-vector field associated with a unit p-chain should containthe corresponding (p-dimensional) mesh element and be con-tained in the union of the d-dimensional mesh elements inter-secting that p-dimensional mesh element; iii) integrability:each smearing field should be square-integrable (and normal-ized in such a way that the integral of its square equals one);iv) asymptotic completeness: in the limit of infinite mesh re-finement, all square-integrable multi-vector field should beapproximated by a linear combination of the smearing fields.

Once a set of suitable smearing fields is constructed, theGram matrix G of the unit chains is obtained by definingthe inner product of any two of them to be equal to theinner product between the corresponding multi-vector fieldsinduced by the Riemannian structure of the underlying man-ifold M. Therefore, G is block-diagonal, all p-vector beingthought of as orthogonal to all q-vectors, for p different fromq; moreover, each block is asymptotically sparse, since theintersection of the supports of most pairs of smearing fields

is negligible in a reasonably fine mesh.

1.1 OverviewIn the following Sections, we consider a cell complex K rep-resenting a finite partition of a Riemannian manifold mod-eled on Ed, i.e. a real differentiable d-dimensional manifoldM in which each tangent space is equipped with an innerproduct (i.e., a positive definite bilinear form), which variessmoothly from point to point. Of course, the Euclideand-dimensional space Ed is itself a (very special) Rieman-nian manifold. The hierarchy of measures (length, area,volume, . . . ) induced by the metric structure of M sur-faces already in Section 2.1, where it is mimicked by mea-sured chains. In Section 6 the notion of metrized chains isintroduced, in order to approximate the metric structure it-self. This allows us to produce a hierarchy of Laplace-deRham operators on cochains that approximate the corre-sponding Laplace-Beltrami operators on differential formsover M (Section 7). Preliminary numerical results for thePoisson equation on 2-dimensional tori endowed with dif-ferent metric structures (flat and curved) are presented anddiscussed in Section 8.

2. CHAINSPreliminary to the introduction of p-chains, we define thep-skeleton Kp ⊂ K of a cell complex K as the subset oforiented p-cells of K, and denote with kp its cardinality.The p-skeleton Kp will be ordered by labeling each p-cell

with a positive integer: Kp = (σ1p , . . . , σ

kpp ).

Let (G,+) be a (nontrivial) abelian group. A p-chain on Kwith coefficients in G is a mapping cp : Kp → G such that,for each σ ∈ Kp, reversing a cell orientation changes the signof the chain value:

cp(−σ) = −cp(σ) .

Chain addition is defined by addition of chain values. Ifc 1p , c

2p are two p-chains, then (c

1p + c

2p )(σ) = c

1p (σ) + c

2p (σ),

for each σ ∈ Kp. The resulting (abelian) group is denotedCp(K;G).

Let σ be an oriented cell in K and g ∈ G. Then, the ele-mentary chain gσ is defined as the chain whose value is gon σ, −g on −σ , and 0 on any other cell in K. Each chainmay then be written in a unique way as a (finite) sum ofelementary chains:

cp =

kpXk=1

gk σkp . (1)

If we take for G the smallest nontrivial group, i.e., G ={−1, 0, 1}, then cells can only be discarded or selected, pos-sibly inverting their orientation.

2.1 Measured ChainsSince we want also to scale chains, we need them to takevalues in a field of scalars F. In fact, mesh refinement (andcoarsening) requires chains to be also scaled, and so addi-tion and subtraction of chains do not suffice. Chains arethen endowed with the additional structure imparted to Fby multiplication. For reasons expounded below, we take

in particular F = R, the real field. Real-valued chains—without an explicit mention of scale and size—were alreadypresent in the 1957’s book by Whitney: in fact, he meant toestablish a geometric foundation for integration.

Real-valued chains may be used to attach a signed p-measureto p-cells, i.e. length to 1-cells, area to 2-cells, volume to 3-cells, and so on, thus restoring part of the geometrical infor-mation stripped away by the purely topological constructionof a cell complex.

In fact, mesh refinement (and coarsening) requires chainsto be also scaled, being representatives of measures, and soaddition and subtraction of chains do not suffice.

Thanks to the multiplicative structure of the real field, allR-valued elementary chains on the p-cell σkp are multiples(or submultiples) of the unit chain ukp, whose value is 1 on

σkp , −1 on −σkp and 0 on any other cell in Kp. Therefore,each chain cp can be seen as a linear combination of unitchains, and Cp(K; R) as a linear space over R, spanned bythe set of unit p-chains, which constitutes its standard basis:

cp =

kpXk=1

λk ukp . (2)

We stress the fact that the representation (1) would formallycoincide with (2), if each cell σkp were identified with the cor-

responding unit chain ukp. However, this seemingly harmlesschoice should definitely be avoided, being equivalent to as-signing one and the same size to all cells. Conversely, wewant to adjust cell sizes, by identifying each cell with a se-lected elementary chain, i.e., a multiple of the correspondingunit chain:

σkp ∼= µkp ukp , (3)

the scalar µkp being the size imparted to the cell σkp . The

size µkp is assumed to be positive—the orientation being ac-

counted for by ukp —and significantly different from zero:

µkp � 0 . 0-cells are systematically given a unit size: µk0 =1 ⇔ σk0 ∼= uk0 . Our motivation is illustrated in the followingexample.

Example 1 (Chain Complex). Three different parti-tions of the same 2-D interval are arranged in Figure 1.The leftmost and rightmost partitions are identical as cellcomplexes, i.e., on mere topological grounds. As chain com-plexes, however, they may differ: their two elementary 2-chains, if meant to represent (cell area)/(overall area), takethe values (.5, .5) for the leftmost mesh, and (.75, .25) forthe rightmost one. The middle mesh is a refinement ofthe leftmost one: its 3 elementary 2-chains take the values(.5, .25, .25).

3. COCHAINSBy definition, the group of p-cochains of K with coefficientsin G is the group of homomorphisms of Cp(K;G) into G:

C p(K;G) := Hom(Cp(K;G), G),

σ22

σ12 σ12

σ22σ32 σ

22

σ12

Figure 1: Three 2-complexes.

i.e., γ p ∈ C p(K;G) if and only if γ p : Cp(K;G)→ G and

γ p“ kpX

k=1

gk σkp

”=

kpXk=1

γ p“gk σ

kp

”for all (g1, . . . , gkp) ∈ Gkp .

If (co)chains are real-valued, then γ p(λup) = λ γp(up), and

each cochain γ p can be seen as a linear combination of theunit p-cochains η p1 , . . . , η

pkp

, whose value is 1 on a unit p-

chain and 0 on all others:

γ p =

kpXk=1

λk η pk , where ηpj (u

ip) =

1 if i = j ;0 otherwise.

(4)

Therefore, the space C p(K; R) can be seen as the dual spaceof Cp(K; R). It is spanned by the set of unit p-cochains,which constitutes the basis dual to the standard basis ofCp(K; R). The evaluation of a real-valued cochain is aptlydenoted as a duality pairing, in order to stress its bilinearproperty:

γ p(cp) = 〈γ p, cp〉.Real-valued cochains represent densities with respect to themeasures imparted to cells by real-valued chains, and theabove duality pairing is a discrete preliminary to integration:its value yields the integral of the density γ p over the chaincp, i.e., the content in the support of cp of the quantitygauged by γ p.

4. BOUNDARY AND COBOUNDARYThe boundary operator

∂p : Cp(K)→ Cp−1(K)

is first defined on simplices (see [21, Ch. 1 § 5]). The nextstep is to extend ∂p to cells by partitioning them into sim-plices, and assuming ∂p to be additive. The boundary oper-ator is then extended to elementary chains, by taking

∂p(gσ) := g(∂pσ) ,

and finally to all chains by additivity. Furthermore, ∂p isassumed to preserve the linear structure real-valued chainshave been endowed with.

The coboundary operator

δ p : C p(K) → C p+1(K)

acts on p-cochains as the dual of the boundary operator ∂p+1on (p+ 1)-chains: for all γ p ∈ C p and cp+1 ∈ Cp+1,

〈δ pγ p, cp+1〉p+1 = 〈γ p, ∂p+1 cp+1〉p .

For once, we have labelled each duality bracket with thedimension of the (co)chains it acts on. Recalling that chain-cochain duality means integration, the reader will recog-nize this defining property as the combinatorial archetypeof Stokes’ theorem. Denoting the dual of an operator bystarring its symbol, we shall write:

δ p = ∂ ∗p+1 .

By definition, the coboundary operator is a homomorphismpreserving the linear structure real-valued cochains have beenendowed with.

5. INCIDENCE & ADJACENCY MATRICESSince p-chains and p-cochains form dual linear spaces, theylend themselves to the usual representation of vectors andcovectors as column and row matrices. The standard basisof Cp and the dual basis of C

p will be used throughout.

The components g1, . . . , gkp of a p-chain cp in (2) may beorganized into a column matrix cp = [cp]. Analogously,the components g1, . . . , gkp of a p-cochain γ p in (4) may beorganized into a row matrix y p = [γ p]. The duality pairingbetween γ p and cp is represented by a matrix product:

〈γ p, cp〉 = [γ p][cp].

Furthermore, we shall represent ∂p by means of the kp−1×kpmatrix [∂p] of its components with respect to the standardbases of Cp(K) and Cp−1(K).

Analogously, we shall represent δ p by means of the kp+1×kpmatrix [δ p] of its components with respect to the standardbases of C p(K) and C p+1(K), i.e., the bases dual to thestandard bases of Cp(K) and Cp+1(K).

Since we use dual bases, matrices representing dual opera-tors are the transpose of each other: for all p = 0, . . . , d−1 ,

[δ p] = [∂p+1]t .

The intersection between p-cells and relatively closed (p+1)-cells may be characterized by the p-incidence matrix Bp,whose entries B ijp are defined as follows:

B ijp = 0 if σip ∩ σjp+1 = ∅ , σ being the closure of σ;

B ijp = ±1 otherwise, with +1 (−1) if the orientationof σip is equal (opposite) to that of the cor-

responding face of σjp+1 .

We now introduce the notion of measured p-incidence, bydefining a matrix Mp whose entries M

ijp depend on both

topology (through p-incidence) and measure (through cellsize):

M ijp := (µip/µ

jp+1)B

ijp .

It should be no surprise that Mp is exactly the matrix rep-resenting the boundary operator ∂p+1 with respect to thestandard bases of Cp+1 and Cp (here and in the following,the complex K is often left implied):

[∂p+1] = Mp . (5)

Consequently, its transpose represents the coboundary op-erator δ p with respect to the dual bases of C p and C p+1:

[δ p] = Mtp . (6)

If size represents length, area, volume for 1-, 2-, and 3-cells,respectively, then measured-incidence matrices have physi-cal dimension (length)−1. Hence, [∂] and [δ] act as first-orderdifference operators.

In graph theory, the adjacency matrix of vertices is one ofthe many representations of a graph, i.e., a 1-complex K ∼=(K0,K1). The well-known relation between incidence andadjacency matrices of a graph can be extended to incidenceand adjacency matrices of all orders p ≤ d of a general d-complex, for any d ∈ N .

Let Mp be the above introduced measured p-incidence ma-trix. By post- and pre-multiplying it by its transpose, youobtain respectively:

A+p := Mp Mtp , A

−p+1 := M

tp Mp . (7)

The (symmetric) matrix A+p is, by definition, the adjacencymatrix between p-chains through (p+1)-chains; analogously,A−p+1 is the adjacency matrix between (p+1)-chains throughp-chains.

6. METRIZED CHAINSAs established by (5) and (6), Mp represents the boundaryoperator ∂p+1 with respect to the standard bases of Cp+1

and Cp, and its transpose Mtp the coboundary operator δ

p

with respect to the dual bases of C p and C p+1. There-fore, their products in (7), while legitimate as matrix op-erations, cannot possibly represent products of boundaryand coboundary operators, unless chains are identified withcochains. This identification is performed by introducing asequence of linear isomorphisms Gp (0 ≤ p ≤ d) betweenchain spaces and their dual cochain spaces:

· · · Cp+1 Cp Cp−1 · · · C1 C0 .

· · · Cp+1 Cp Cp−1 · · · C1 C0

Gp+1 Gp Gp−1 G1 G0

∂p+2 ∂p+1 ∂p ∂p−1 ∂1

δp+1 δp δp−1 δp−2 δ0

A non-degenerate bilinear form gp : Cp × Cp → R is as-sociated with each isomorphism Gp, thus establishing aninner-product structure on the space of p-chains:

gp(cip, c

jp ) := 〈Gp cip , c jp 〉 . (8)

The inner product gp is symmetric if and only if the iso-morphism Gp is self-dual, i.e., G

∗p = Gp. The inverse of the

dual

G p := (G∗p)−1

endows the space of p-cochains with the inner productgp : C p × C p → R defined analogously to Equation (8):

gp(γ pi , γpj ) := 〈G

p γpi , γpj 〉 = 〈γ

pi , G

−1p γ

pj 〉 , (9)

so that gp(Gp cip, Gp c

jp ) = gp(c

ip, c

jp ) holds identically.

The isomorphism Gp and the associated bilinear form gpare represented by the Gram matrix Gp, whose entries G

ijp

are the components of the Gp-images of the elements of the

standard basis of Cp in the dual basis of Cp:

Gp uip =

kpXj=1

G ijp ηpj ;

equivalently,

G ijp := gp(uip, u

jp) .

It is easily seen that the Gram matrix Gp representing G p

and gp is the inverse transpose of Gp:

Gp = G−tp .

The trivial inner product is defined by the assumption:

Gp = Gp = I kp×kp ⇔ Gp u

ip = η

pi . (10)

It makes the standard bases of Cp and Cp orthonormal,

identifying each unit chain with the corresponding unit co-chain. Assumption (10), while expedient to use on any givenK, is totally unrelated—in general—to the geometric prop-erties relevant to the physical phenomena taking place on theunderlying manifold M. Dealing properly with the identifi-cation between chains and cochains is essential for importinginto the discrete model the relevant, physics-based metricstructure. This issue is also basic to gain the possibility of ameaningful information transfer from a cell complex to anyof its refinements (and vice versa), and to establish a notionof convergence for refinement sequences.

To bridge the metric gap between the Riemannian manifoldM and the cell complex K supposed to approximate it, weassociate to each unit p-chain a p-vector field on M (thenotion of tangent vector and multi-vector fields on a differ-entiable manifold being taken for granted). By doing so, weestablish a linear mapping of the space of p-chains into thespace of p-vector fields:

Sp : Cp(K)→ Xp(M) , (11)

which we call p-smearing map. Let us call unit p-vectorfields the images of unit p-chains under Sp. The support ofeach unit p-vector field Sp up contains the corresponding p-dimensional mesh element and is contained in the (closure ofthe) union of the d-dimensional (d=dimM) mesh elementsintersecting that p-dimensional mesh element. Secondly, theunit p-vector field Sp up is tangent to the correspondingmesh element. Thirdly, it is square-integrable (and normal-ized in such a way that the integral of its square equals one).Obviously, these criteria may be satisfied in infinitely manyways (but all reasonable choices should be asymptoticallyequivalent—with varying rates of convergence, though). InSection 8 we explore the simplest possible construction, basedon piecewise constant p-vector fields.

Once the p-smearing map Sp is selected, the Gram matrixGp is computed through integration on M:

G ijp =

ZM

`Sp u

ip

´(x)·

`Sp u

jp

´(x) dV (x) , (12)

where the inner product between p-vectors tangent to Mat x ∈ M is denoted as a dot product, and dV (x) is thed-volume form induced by the Riemannian metric on TxM.Clearly, the properties of the dot product are inherited bygp, which turns out to be symmetric and positive definite.

Hence, Gp is self-dual and Gp = G−1p . Moreover, since for

a reasonable fine mesh the intersection of the supports ofmost unit p-vector fields is negligible, Gp is sparse.

7. LAPLACE-DE RHAM OPERATORSThe adjoint (or transpose) ∂>p+1 of the boundary operator∂p+1 is characterized by the property:

Gp`∂p+1cp+1, cp

´= Gp+1

`cp+1, ∂

>p+1cp

´,

to be satisfied for all cp ∈ Cp and cp+1 ∈ Cp+1. It is easilychecked that

∂>p+1 = Gp+1 ∂ ∗p+1Gp .

Consequently (recall Equation (6)),

[∂>p+1] = Gp+1Mtp Gp .

Therefore, the (symmetric) adjacency operators

∂p+1∂>p+1 , ∂

>p+1∂p+1 (13)

are represented respectively by the matrices

Mp Gp+1 Mtp Gp , G

p+1 Mtp Gp Mp .

The discrete Laplace-de Rham operators—a fundamental andubiquitous ingredient of physical modeling—are defined assums of duals of adjacency operators: for all 0 ≤ p ≤ d ,

∆p :=`∂p+1∂

>p+1 + ∂

>p ∂p

´∗ = (δp)>δp +δp−1 (δp−1)>, (14)it being intended that ∂0 and δ

d (and hence (δ0)>and ∂>d ) arenull. The Laplacian on p-cochains is therefore representedby the matrix

[∆p] = Gp Mp Gp+1 Mtp + M

tp−1 Gp−1 Mp−1 G

p.

Hence, the straightforward representation

[∆p] = A+p + A

−p

is only valid if the inner product hierarchy is trivial (recallEquation (7)).

8. NUMERICAL EXAMPLESIn the following we will show some numerical examples ofour metric-aware approach, solving the following ellipticalproblem, where u is considered a scalar field:

∆u = f , (15)

We shall consider a toroidal domain D = [0, 2π) × [0, 2π)discretized using square elements. The choice of a toroidaldomain, i.e. with periodical boundary, allows us to disregardboundary conditions obtaining a unique solution of Equa-tion (15), except for an additive constant. Our exampleswill therefore consider the reference manifold as D ≡ [[K]],considering different metrical properties on D defined by thechosen inner product hierarchy. In particular, we will com-pare the solution of Equation (15) on the domain D dis-cretized using regular orthogonal grids, with the solutionon a 45 degrees inclined grid, as pictured in Figure 2; nextwe we will furnish the same toroidal domain D with a Rie-mannian metric. Common sense usually discourages suchdistorted meshes, and in the following we show that em-ploying a metric-aware approach we may promptly recoverthe accuracy usually lost with highly deformed cells.

Figure 2: Depiction of 1-vector fields S1u1 in red,and S1u

′1 in blue. Note that the vector fields, whose

direction are represented by arrows, are not orthog-onal on the distorted mesh (right).

In the following, our choice of smearing maps Sp yields,for any up ∈ Cp(K), a constant vector field on D with thesame orientation as up, as pictured in Figure 2, normalizedover the domain, i.e., that the integral of over the domain〈Spup, Spup〉 produces a unitary scalar value.

We highlight the fact that in these examples, 0- and 2-chainshave orthonormal bases, and therefore G0 and G2 are diago-nal matrixes. On the other hand, 1-chains have an orthonor-mal base if, and only if, the underlying grid is orthogonal,as depicted in Figure 2.

Example 2 (Harmonic function). Let us consideras known field f defined on the domain D the harmonicfunction f(x1, x2) := − sin(x1). The elliptical problem de-scribed in Equation (15) was solved numerically on both theorthogonal and inclined grids.

Figure 3: A section of the solution on an orthogonalgrid (blue), on the 45 degrees distorted mesh (red),and the exact solution (purple).

Our results, pictured in Figure 3 along with the exact sym-bolic solution (shown in purple), reveal that employing ourmetric-aware approach we obtain appreciable results in pres-ence of significant mesh distortion (red), with the orthogonalone (shown in blue), sampled at x2 = π/2.

Example 3 (Dirac function). In this example, ourEquation (15) is solved for f(x1, x2) :=u0(x1 − π/4, x2 −π/4)−u0(x1−3π/4, x2−π/4), where u0 represents the diracfunction. Figure 4 shows the graph of our solution (top), and

a section for both orthogonal and inclined grids, along withthe graph (bottom).

Figure 4: The graph of the solution of Example 3(top), and a section (bottom): solution on the or-thogonal grid (blue), on the 45 degrees distortedmesh (red).

We are now promoting the domain D, a plain torus, to theRiemannian torus S1 × S1. In the following examples theundelying topology stays untouched, while all masure- andmetric-related features reflect the geometric properties im-posed by the new domain. In particular, we solved Equa-tion (15) reflecting both Examples 2 and 3 on two torusesT1 and T2 with different radii, in order to exemplify the in-fluence of pure metrical properties on the results. In detail,T1 has radii 3.5 and 2.5, while T2 is generated by two circleswith radii 5.95 and 0.05: in these example T2 is thereforevirtually monodimensional, while T1 retains the riemannianmetrical structure. Additionally, in order to achieve a betteraccuracy, in Example 5 will be carried on a locally-refinedmesh on T1 (see Figure 7).

Example 4 (Harmonic function on S1 × S1). Weare now transferring the problem illustrated in Example 2from a plain torus to the analogous Riemannian manifold,

or in other words, we are considering the same domain em-ploying another particular choice for the inner product onchains.

In particular, we solve the equation ∆u = − sin(x1) on T1and T2, with the results pictured in Figure 5. In details wecan see that the virtually monodimensional torus T2 yieldsa solution that is not influenced by the metrical structure,while a substantial deformation is present on T1, as picturedby the graph of the solution on the bidimensional unfoldingof our domain (see Figure 5, middle).

Example 5 (Dirac function on S1 × S1). In thisexample we transpose Example 3 on our two Riemanniantoruses. Since no symbolic solution may be given for Equa-tion (15) on our domain, we compare our results with a nu-merical one employing a third-party commercial software. Inthis example, in order to achieve a higher accuracy in oursolution, we performed a local refinement of our cell complexin proximity of our singularities.

Figure 6 shows a section of the solution on both T1 and T2:our results were compared with the one provided by the third-party software, to be considered as “exact” (on the left, shownin red). We notice as the riemannian metric has virtually noinfluence on T2; on T1 our solution differs from the referencenumerical result, while achieving better accuracy upon a localrefinement.

The solution on T1 is picured in Figure 7 along with theresults obtained with a local mesh refinement.

9. FINAL REMARKSIn this paper we establish a link between Riemannian mani-folds and their discrete counterpart represented by cell com-plexes, integrating measure-theoretical properties and metri-cal information within the general algebraic-topological con-cept of chain complexes.

By integrating the standard notion of chain complexes withmeasure-theoretical concepts, we aim at recovering the geo-metrical information stripped away by a classic topologicalapproach. Moreover, we define a family of inner products Gpon p-chains that, in addition to measure, recovers also theRiemannian metric by associating a p-vector field to eachunit p-chain.

We therefore define a link between chains and cochains thatreflect the underlying Riemannian manifold, and, by con-sidering chains of all dimensions, we construct the completehierarchy of Laplace-de Rham operators on cochains, i.e. the(discrete) counterpart of the Laplace-Beltrami operators ondifferential forms.

We mention in passing that the p-vector fields associated top-chains play an essential role not only in the definition ofan inner product on chains, but also in the construction ofa wedge product of chains via integration on the underlyingmanifold. Further study should be devoted to understandingthe asymptotic structure of the inverse of Gp, i.e., of theinduced inner product on cochains, and how to constructlocal approximations to it.

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Figure 5: Solution on T1 of Example 4 (left), and its bidimensional unfolding for T1 (right) and T2.

Figure 6: Sections of our solution (middle) of Example 5 on T2 (left) and T1 (right), see Figure 7. On theleft, our solution is compared with a reference numerical result (in red); a local refinement on T1 is picuredin green (right).

Figure 7: Solution on T1 of Example 5 on a regular grid (left) and employing a local refinement (middle)detailed on the rightmost picture.

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