A Piecewise Constant Interpolation of dec

Remark. Note that the following situations are ruled out by our assumptions:

<a> an m-dimensional sphere with V ={p0, . . . , pm}, because BV would consist of two points of equidistance, which is not a0-ball, but a0-disk. However, the denition of Voronoi regions would be feasible, but its dual would consist of two m-simplices with the same vertices, and our notation does not allow to distinguish between them.

<b> m+ 2 equidistant points V = {p0, . . . , p_m+1} in an m-dimensional manifold, because thisV is not generic: In fact,B_V would be the point of equidistance, but this set should be empty, as|V|> m+ 1.

<c> The counterexample of Boissonnat et al. (2011, pp. 38sqq.), because the bisector B_{p,u,v,w} is not empty.

8.7 Denition. Situation as in 8.1 with genericV and2δ <cvrM. LetKbe the complex from 8.5. Fore∈Kⁿ, let x_e be the mapping from 5.4. As x_e|_rf only depends on ffor f⊂e, this piecewise denition gives a well-dened mapping x: rK→ M, called the KarcherDelaunay triangulation ofM with vertex setV.

8.8 Proposition. The KarcherDelaunay triangulation is indeed a triangulation in the usual sense: Situation as in 8.1 with genericV and2δ <cvrM. Ifδ is so small that the requirements of 6.19 are met on each Karcher simplex, thenxis bijective.

Proof. The mapxis surjective because its image is non-empty and has no boundary inM. By 6.19 eachxeis injective, and as the Karcher simplices do not overlap except

on their boundaries, so isx, q. e. d.

8.9 Remark. <a> IfM is not closed, but compact and with boundary, the construction is of course feasible, but will only be bijective if there are also points on the boundary and the boundary is aligned with their KarcherDelaunay triangulation.

<b> The construction of Cheeger et al. (1984) seems similar, but (of course) does not use our barycentric mapping. It starts with a triangulationx:rK→M, considers ner and ner subdivisionss :rK⁰ →rK of the complex, and then compares the metric(s◦x)^∗g onrK⁰ to the piecewise at metric induced by edge lengths`ij = d(x◦s(ri), x◦s(rj)), for edges ij∈(K⁰)¹ in the subdivided complex.

<c> We know, however, that Burago et al. (2013) state that it is now clear that in di

mensions beyond three polyhedral structures are too rigid to serve as discrete models of Riemannian spaces with curvature bounds, but nevertheless there will certainly be rigidity results for spaces of piecewise constant curvature without counterparts in the smooth cate

gory, we are not convinced that the references they give support this statement in its full generality.

9. A Piecewise Constant Interpolation of dec

Goal. As second main construction of this thesis, we will now give an interpretation of the discrete exterior calculus as piecewise constant dierential forms, which turns variational problems in the simplicial cohomology(C^k, ∂^∗)into problems in a complex

(P⁻¹Ω^k, d). The main question will be the connection betweend and the usual exte

rior derivativedonH^1,0Ω^k. The introductional denitions are the basics of simplicial homology as they can be found in any topology textbook, e. g. Munkres (1984) or Hatcher (2001).

Discrete Exterior Calculus (dec)

9.1 Denition. LetRbe a ring with neutrals0and1, and letKorbe a regularn -dimensio-nal oriented simplicial complex. For any simplex s∈K^k_or, let χs:K^k_or→Rbe dened byχs(s) = 1andχs(s⁰) = 0for anys⁰ 6=s.

Consider the R-module C˜k(K) that is spanned by all χs, s∈ K^k_or. Let Ck(K), the space of k-chains overK(with coecients inR), be the quotient ofC˜k(K)under the identication of χ_s⁻ and −χs. Its dual spaceC^k(K), the R-module of all homomor

phismsCk(K)→R, is called the space of k-cochains over K(with coecients inR).

Letf^sbe the generators of C^k(K)dual to δ_s, that meansf^s(χ_s) = 1andf^s(χ_s⁰) = 0 fors6=s⁰. In the following, we will only use R=R.

The boundary operator is the linear map C_k(K) → C_k−1(K), dened on the generators by

∂ χ_[p0,...,pk]:= (−1)ⁱχ_{[p0,...,ˆpi,...,pk]}

(as usual, summation over iis intended), wherepˆi means that this vertex is omitted.

With respect to the basisχs, we write∂ in coecients:

∂ χs=∂_s^tχt fors∈K^k,

where summation overt∈K^k−1is intended. For the whole section, we will sum about indices occuring twice in a product, irrespective if they are superscripts oder subscripts.

Volume terms like|s| or |U(s)| do not count for this, assis no sub- or superscript in them. The co-boundary operator∂^∗is the dual of∂, i. e. a mapC^k−1(K)→C^k(K) uniquely characterised by ∂^∗α(c) =α(∂c)for allα∈C^k−1(K)and allc∈C_k(K). 9.2 Remark. <a> By a direct computation, or by common linear algebra knowledge, one

obtains that the matrix representation of∂^∗is the transposed of the matrix represen

tation of∂. In other words,(∂^∗)^t_s=∂_s^t for∂^∗f^t= (∂^∗)^t_sf^s.

<b> It would be very natural to write δs instead of χs, because χs actually is the Kronecker delta onK^k. But we will already have some operatorδacting on dierential forms, we will dene some δfor cochains and someδ on piecewise constant forms. In the whole following section, we will not use the Kronecker symbol.

<c> The use of functions χs as generators of Ck(K) is only one possible denition. The other frequently encountered possibility is to speak of formal linear combinations of thes themselves (e. g. Hatcher 2001 and Hirani 2003 use this denition). Logically, there is no dierence between both denitions, as the only strict way to dene formal linear combina

tions is to use the characteristic functions χs. However, the existence of both approaches introduces an unpleasant notational ambiguity that may disturb a quick reader: Linear maps from simplices toRare chains in our notation, whereas they represent cochains in the other.

Our notation has the advantage to employsonly as sub- and superscript, but not as term, which allows for usual summation convention.

9. A Piecewise Constant Interpolation of dec 9.3 Lemma. Let K consist of one single n-simplex. Then the boundary map ∂_k : C_k → C_k−1, which can be written as ⁿ⁺¹_k+1

× ⁿ⁺¹_k

-matrix, has rank ⁿ_k .

Proof. The matrix size just stems from counting the elements inK^k, which arise from choosingk+ 1vertices out ofn+ 1.

The rank of∂kis proven by induction overk, starting withk=n. Here the statement is that∂n :Cn→Cn−1has rank one, which is true because∂n6= 0. For anyk < n, the rank-nullity theorem gives that the rank of∂_k can be computed as dimension ⁿ⁺¹_k+1 its image space minus the dimension of its kernel, which is the rank of ∂_k+1 becauseof thek'th homology group of the simplex vanishes. And by assumption,∂_k+1 has rank

n k+1

, which gives rank∂_k= ⁿ⁺¹_k+1

− _k+1ⁿ

= ⁿ_k, q. e. d.

9.4 Short introduction to discrete exterior calculus (dec). The discrete exterior calculus (Desbrun et al. 2005, Hirani 2003) attempts to build a simple and useable nite-dimensional version of the de Rham cohomology based on an intelligent interpre

tation of simplical cohomology. It calls∂^∗the discrete exterior derivatived, which gives thatd:C^k(K)→C^k+1(K)acts as

df^t =d^t_sf^s withd^t_s=∂^t_s. (9.4a) If pointsλs ∈ rsfor all simplices s and numbers ak ∈ R are given, leading to dual cellsr(∗s)as in 4.16a, it denes the scalar product of two discretek-forms as

hhhαsf^s, βs⁰f^s0iii_Ck:=akαsβs

|∗s|

|s|. (9.4b)

Remark. The numbersa_k do usually not occur in the denition of the scalar product, but we will see that they must be chosen asa_k = ⁿ_k

to obtain a correspondence to piecewise constant forms. The pointsλ_sare classically chosen to be the circumcentres of thers.

The coderivativeδis supposed to be dual to dwith respect to this scalar product, that meanshhhα, dβiii_Ck =hhhδα, βiii_Ck−1 for allα∈C^k(K)and allβ ∈C^k−1(K). Spelling out both sides forα=f^s andβ=f^t gives

d^t_s^|∗s|_|s| a_k =δ_t^s|∗t|_|t|a_k−1, ⇔ δ^s_t = a_k ak−1

|∗s| |t|

|∗t| |s|∂^t_s. (9.4c) Other denitions are obvious: A formαis called harmonic if(δd+dδ)α= 0etc.

Piecewise Constant Differential Forms

9.5 Situation. Let Kor be an oriented regular n-dimensional simplicial complex with a discrete Riemannian metricg, let Kbe the corresponding non-oriented complex, and letλsfor any simplex sdene subdivision neighbourhoodsU(s).

9.6 Denition. Situation as in 9.5. LetP⁰Ω^kbe the space ofL^∞Ω^kforms that are constant inU(s)for eachs∈K^k.

Any simplexs∈K^k has a volumek-formdvol_rswhich can be extended to a constant k-form in wholeU(s). Denoting the extension also asdvol_rs, let

ω^s:=

®dvolrs inU(s)

0 elsewhere, and P⁻¹Ω^k:= span{ω^s:s∈K^k} ⊂P⁰Ω^k. 9.7 Example. Situation as in 9.5, dimensionn= 2. Consider two trianglesrijkandrjil,

which together contain the subdivision neighbourhoodU(ij). Thenω^ij is the attened unit vector vector eld in direction ri−rj in U(ij) and zero elsewhere, ωⁱ is the characteristic function ofU(i), and similarlyω^ijk is the volume form ofrKinrijkand the zero2-form elsewhere.

9.8 Observation. All basis elements have pointwise unit length with respect to the metric induced on the tensor bundles byg, and have distinct support up to null sets, so the L² scalar product has diagonal form in the basisω^s:

hhhαsω^s, βs⁰ω^s0iii_L2Ω^k =|U(s)|αsβs (9.8a) Denition. Situation as in 9.5. Letd:P⁻¹Ω^k−1→P⁻¹Ω^k be dened by

dω^t=d^t_sω^s, d^t_s:= |t|

|s|∂_s^t. (9.8b)

Letδ:P⁻¹Ω^k →P⁻¹Ω^k−1 be dened by

δω^s=δ^s_tω^t, δ^s_t =|U(s)|

|U(t)|d^t_s. (9.8c)

9.9 Proposition. Situation as in 9.5. The mapsd andδfulll the discrete Stokes' formula

ˆ

rs

dα= ˆ

∂rs

α, (9.9a)

discrete Green's formula hhhdα, βiii_L2Ω^k=hhhα, δβiii_L2Ω^k−1 (9.9b) for alls∈K^k_or,α∈P⁻¹Ω^k−1, andβ ∈P⁻¹Ω^k. In particular,d²= 0.

Proof. ad primum: For any α∈ Ω^k−1, we have ´

∂rs

α=∂_s^t0 ´

rt⁰

α. Now letα =ω^t for some t∈K^k−1. Then we have

ˆ

∂rs

ω^t =∂_s^t0 ˆ

rt⁰

ω^t=∂_s^t|t|

(without summation overt). On the other hand, ˆ

rs

dω^t= ˆ

rs

d^t_s0ω^s0 =d^t_s|s|.

ad sec.: If one spells out both scalar products with help of 9.8a forα=ω^t∈P⁻¹Ω^k−1 andβ =ω^s∈P⁻¹Ω^k, one getsδ_t^s|U(t)|=^! d^t_s|U(s)| for 9.9b to hold, q. e. d.

9. A Piecewise Constant Interpolation of dec 9.10 Remark. The discrete Green's formula 9.9b holds without assumption on the bound

ary values because the weights|U(s)|and|U(t)|already incorporate the smaller extent ofδ. For a correct treatment of boundary conditions in variational problems, one would have to modify 9.8c. We decided to investigate the original dec setup here.

9.11 Proposition. Situation as in 9.5. The mapi_k:C^k →P⁻¹Ω^k,f^s7→ _|s|¹ω^sis a cochain map, i. e. each square in the following diagram commutes:

C⁰

d - C¹

d - . . .

d - Cⁿ

PΩ⁰ i₀

? d

- PΩ¹ i₁

? d

- . . . d

- PΩⁿ i_n

?

If a_k = ⁿ_k, it is an isometry for each k and a chain map, i. e. each square in the following diagram commutes:

C⁰

δ C¹

δ . . .

δ Cⁿ

PΩ⁰ i₀

?

δ PΩ¹ i₁

?

δ . . .

δ PΩⁿ i_n

?

Proof. The isometry property is clear by the expressions 9.4b and 9.8a for the scalar product ofC^k and P⁻¹Ω^k respectively. The propertiesdi_k−1 =i_kdand δi_k =i_k−1δ only need to be checked for basis elements, so it suces to show

d^t_s|t|¹ =d^t_s|s|¹, δ_t^s_|s|¹ =δ_t^s_|t|¹ for alls∈K^k,t∈K^k−1.

The rst one is obvious from denitions 9.4a and 9.8b. The second one comes from 9.4b, as

δ^s_t =|t| |U(s)|

|s| |U(t)|∂^s_t =

n k

n k−1

|∗s|

|∗t|∂_t^s= ak−1

ak n k

n k−1

|s|

|t|δ_s^s,

q. e. d.

Remark. It might seem a little bit queer to use piecewise constant forms for this construction and not the elementary forms introduced by Whitney (1957, sec. IV.27)

˜

ω^[p0...pk]=k!λⁱdλ0∧ · · ·dλ”ⁱ· · · ∧dλ^k,

which would also makeia cochain map. The reason is that we did not succeed to nd any relation between theL² scalar product of Whitney's elementary forms and the dec scalar product 9.4b. This means that although there is a worked-out interpolation estimate for the space spanned byω˜^s,s∈K^k, by Dodziuk (1976), it gives no possibility to compare solutions of variational problems that were computed using the dec scalar product.

9.12 Proposition. Suppose thatrKg is a piecewise at, (ϑ, h)-small and absolutely well-centred realised simplicial complex, that means all circumcentres λ_s have barycentric

coordinates λⁱ_s> α, and that the circumradii are bounded byβh. Then ifg¯is a second piecewise at metric with|(g−g)hv, wi| ≤¯ ch²|v||w|, it holds forc⁰:= _αϑ^cβ:

ω^s_g−ω_g^s_¯L2.c⁰h² ω^s_gL2

dgω^s_g−d¯gω_g^s_¯L².c⁰h² dgω_g^sL²

|hhhα_sω_g^s, β_s⁰ω_g^s0iii_g− hhhα_sω^s_¯g, β_s⁰ω^s_¯g⁰iii_g¯|.c⁰h²hhhα_sω^s_g, β_s⁰ω^s_g⁰iii_g

Proof. The dierence between d_g and d_¯g is easiest, because it only involves simplex volumes like|s|_gand|s|_¯g. These are close to each other by 3.20. The approximation of the scalar product involves comparison between the neighbourhood volumes |U_g(s)|_g and|Ug¯(s)|¯g. These can be estimated if we know how the circumcentres are distorted.

By 3.12a, these are controlled by the distortion of the CayleyMenger matrix inverse M₊⁻¹, and inverses of symmetric matrices are treated by 3.21 (which we apply toM+

instead ofg):

|qⁱ−q¯ⁱ|.ch²r|grad_gλⁱ|

(where r is the circumradius with respect to g) because 4r² and |vⁱ|² = |grad_gλⁱ|² are the corresponding diagonal entries of M₊⁻¹. By assumption, this is smaller than

c⁰h²|qⁱ|, q. e. d.

Connection to the bv Derivative

Goal. Recall that piecewise constant functions possess distributional derivatives, which are (n−1)-dimensional measures concentrated on the jump sets. Their analogue for dierential forms are the currents from geometric measure theory. (In order to avoid currential derivative or similar terms, we will speak of bv derivatives.) If our deni

tion of discrete exterior derivatives is meaningful, it should be connected to this sort of derivative. In fact, the bv derivative of a piecewise constant k-form αalso fullls Stokes' theorem if the jump set is transversal to the integration domain. But as their support is(n−1)-dimensional, we will see that its scaling behaviour does match the one of full-dimensional(k+ 1)-forms such as dα.

9.13 Denition. The comass of ak-covectorαis the absolute value of its largest compo

nent, equivalently: the maximum over all applications ofαto simple unitk-vectors:

||α||∗= maxα(e_i0∧ · · · ∧e_ik).

For completeness, we also dene that the mass of a k-vector is the norm dual to the comass: ||v||∗ = max_||α||∗=1α(v). A dierential form α ∈ L¹_locΩ^k(M) has locally bounded variation (is locally of bv) if

sup

β∈C¹₀Ω^k+1(U)

||β||∗≤1

hhhα, δβiii is nite for allU ⊂⊂M, (9.13a)

where of courseC¹₀Ω^k(U)denotes the space of continuously dierentiablek-forms onM with compact support inside U. The space ofk-forms with locally bounded variation is called BV_locΩ^k. The globalisation to the spaceBVΩ^k is as usual.

9. A Piecewise Constant Interpolation of dec 9.14 Fact (cf. Evans and Gariepy 1992, thm. 5.1). For each α ∈ BV_locΩ^k, there is a Borel-regular measureµonM and aµ-integrable (k+ 1)-formd^BVαsuch that

hhhα, δβiii= ˆ

M

hd^BVα, βidµ for allβ ∈C¹₀Ω^k. (9.14a) We will mostly writehhhd^BVα, βiiias abbreviation of the right-hand side.

Remark. This formulation of the bv structure theorem is the one normally used for functions of bounded variation. For dierential forms, one calls the supremum in 9.13a the mass of the current (linear form on Ω^k) β 7→ hhhα, δβiii, and then observes that every current of nite mass is representable by integration in the meaning of the theorem (Federer 1969, sec. 4.1.7, or Morgan 2000, sec. 4.3b). To obtain uniqueness ofd^BVα, one usually requires it to have unit-mass everywhere, and the pointwise scaling then comes fromµ. As we are only interested ind^BVαforα∈P⁻¹Ω^k, it will be more adequate to use a non-unit-length(k+ 1)-form and the volume form of∂U(s),s∈K^k, forµ.

For the proof of 9.14, we refer to Evans and Gariepy (loc. cit.), because it only consists of the observation that β 7→ hhhα, δβiii has a norm-preserving continuation to C⁰₀Ω^k, and the application of Riesz' representation theorem.

9.15 Proposition. For the basis elementsω^t of P⁻¹Ω^k, the bv derivative is given byµ= dvol∂U(t) andd^BVω^t =ν∧ω^t, whereν is the outer normal on U(t).

Proof. Ifβ ∈ C¹₀Ω^k+1, the product hω^t, δβi is supported only in U(t), where we can apply the classical Green's formula because the integrand is smooth. So

hhhd^BVω^t, βiii^(9.14a)= hhhω^t, δβiii= ˆ

∂U(t)

ω^t∧ ∗β+hhhdω^t, βiii= ˆ

∂U(t)

hν∧ω^t, βidvol_∂U(t), (9.15a) the last equality by usual multilinear algebra anddω^t= 0almost everywhere, q. e. d.

9.16 Proposition. There is a variant of Stokes' theorem for the bv derivative of P⁻¹

forms: If we dene ˆ

rs

d^BVω^t:=

ˆ

rs∩∂U(t)

ω^t, (9.16a)

then ˆ

rs

d^BVα= ˆ

∂rs

α for allα∈P⁻¹Ω^k,s∈K^k. (9.16b) Proof. The homotopy formula is easy for constant forms: Ifdα= 0, then0 =´

Udα=

´

∂Uα, hence ´

Aα = ±´

Bα if the integration domains A and B bound a common (k+ 1)-dimensional domainU, the sign depending on the orientation ofB. This is the case for

∂(rs∩U(t)) = ∂rs∩U(t) ∪ rs∩∂U(t).

So the formula is clear forα=ω^t by denition of the domain integral overrs, and by linearity, it hence holds for allα∈P⁻¹Ω^k, q. e. d.

Remark. <a> The notation 9.16a might seem to obscure the actual integration pro

cess over a subdomain, but we would like to emphasise the analogue to dα, which fullls the same Stokes formula.

<b> For a smoothly bounded(k+1)-dimensional integration domainU and dierential formsα∈Ω^k, β∈Ω^k−1, it is always true that

ˆ

U

α= ˆ

U

hα,dvol_Uidvol_U, ˆ

∂U

β =± ˆ

∂U

hν∧β,dvol_Uidvol_∂U,

where the sign is the same as in dvolU =±ν∧dvol∂U. Therefore, the notation 9.16a can also be interpreted as

ˆ

rs

d^BVα= ˆ

rs∩∂U(t)

hd^BVα,dvolrsidvol_rs∩∂U(t)

withd^BVω^t =ν∧ω^tas in 9.15. The notational problem is mainly that the bv derivative is supported on a codimension-1-set, which makes the integral in Green's formula (n−1)-dimensional instead ofn-dimensional, and the left-hand side integral in Stokes' formulak-dimensional instead of(k+ 1)-dimensional. Unfortunately, we do not know a common notation covering both.

<c> The formula stays correct (with an appropriate notational adaption) for sbv forms (introduced by de Giorgi and Ambrosio 1988, as overview we refer to Ambrosio et al. 2000) which are bv forms whose derivative measureµconsists of partsµ_acandµ_s which are absolutely continuous with respect to then-dimensional and to the(n−1) -dimensional Hausdor measure inrK, ifµ_sis supported on a set that is transversal to the integration domainU. For the proof, one can use the approximation ofα∈SBVΩ^k by convolution with smooth Gaussian kernels. If the jump set ofα, i. e. the support of µ_s, is transveral toU, then the convergence is uniform almost everywhere on∂U, and so the integrals ´

∂Uαi of the mollied formsαi tend to ´

∂Uαand give a well-dened interpretation of´

Ud^BVα.

<d> This means that forβ∈L^∞Ω^k+1 which is smooth inside eachU(s),s∈K^k+1, we have

hhhα, δβiii=X

s

ˆ

∂U(s)

α∧ ∗β+hhhd^BVα, βiii for allα∈P⁻¹Ω^k. 9.17 Proposition. hhhd^BVα, βiii= _k+1ⁿ hhhdα, βiiifor allα∈P⁻¹Ω^k and allβ ∈P⁰Ω^k+1.

Proof. Due to 9.15a, it suces to consider, for eachs∈K^k+1 and eachs∈K^k, hhhd^BVω^t, βiii_U(s)=

ˆ

U(s)∩∂U(t)

ω^t∧ ∗β

which can be spelled out by using the(n−1)-agsa inU(s)∩∂U(t): ˆ

U(s)∩∂U(t)

ω^t∧ ∗β=X

a

ˆ

∆ⁿ⁻¹

ω^t∧ ∗β= 1 (n−1)!

X

a

(ω^t∧ ∗β)(ba),

9. A Piecewise Constant Interpolation of dec whereb_a is the pull-back of an orthonormal basis ofr⁰a. The ags occuring in this sum are of the form(h0i, . . . ,ˆt,s, . . . ,hni), cf. 4.17c. Using the vectorsv_hii,hi+1ifrom 4.18, we have inside eachr⁰a

ba=v_h0i,h1i∧ · · · ∧ v_hk−1i,s ∧v_s,hk+2i∧ · · · ∧v_hn−1i,hni

=v_h0i,h1i∧ · · · ∧(v_hk−1i,t+vt,s)∧v_s,hk+2i∧ · · · ∧v_hn−1i,hni, the factors in the latter product are all mutually perpendicular. Now observe

(ω^t)^]= v_h0i,h1i∧ · · · ∧v_hk−1i,t

|v_h0i,h1i∧ · · · ∧v_hk−1i,t|, (∗ω^s)^]= v_s,hk+2i∧ · · · ∧v_hn−1i,hni

|v_s,hk+2i∧ · · · ∧v_hn−1i,hni|. By orthogonality of all vectors inba, the application(ω^t∧ ∗β)(ba), usually comprising all permutations of the factors, splits as

(ω^t∧ ∗β)(ba) =ω^t(v_h0i,h1i∧ · · · ∧v_hk−1i,t) (∗β)(v_s,hk+2i∧ · · · ∧v_hn−1i,hni)

=|v_h0i,h1i∧ · · · ∧v_hk−1i,t| hβ, ω^si|v_s,hk+2i∧ · · · ∧v_hn−1i,hni|.

Summation over all ags(h0i, . . . ,ˆt,s, . . . ,hni)then gives, by 4.19, ˆ

U(s)∩∂U(t)

ω^t∧ ∗β= k!(n−k−1)!

(n−1)! |t| |∗s|hω^s, βi= n k+ 1

|t|

|s||U(s)|hω^s, βi,

q. e. d.

Approximation Estimate for P⁻¹Ω^k

9.18 Lemma. Let us denote the set of multiindices I = (i₁, . . . , i_k) with i_j < i_j+1 and 1≤i_j ≤n for allj, by ⁿ_k

(which in fact is its cardinality). SupposeKis a simplicial complex with only onen-simplex e with a non-degenerate at metric. Then the ⁿ_k

×

n+1 k+1

matrix

M(k)(e) :=ˆ

re

ω^t(vI)

I∈(ⁿ_k),t∈K^k

has full rank ⁿ_k, wherev_{i1,...,ik}:=Vvi_j for an arbitrary basis vj ofT re.

Proof. The choice ofv_j does not matter, because a change of this basis only results in a multiplication with a non-singular ⁿ_k

× ⁿ_k-matrix from the left. Furthermore, it suces to show that the matrixM˜(k) := (´

dvolrt(vI))I,t has full rank, because it only diers from M_(k) by factors depending on the volumes |U(t)|, which must be non-vanishing for at least ⁿ_k

of the t (which happens if the circumcentre lies on a facet of t). Now if we choose v_i = ri−r0, we can transform the situation onto the unit simplex D with vertices0, e₁, . . . , e_n, where then v_i =e_i, and the volume forms of simplicestcontaining the vertex0 have a particularly easy expression:

dvol_rt=dxⁱ¹∧ · · · ∧dx^ik fort={0, i1, . . . , i_k}.

So dvol_r({0}∪I)(vI⁰) = 1 if I = I⁰ and 0 else, hence these rows of M˜(k) are already

linearly independent, q. e. d.

9.19 Proposition. SupposerKis a simplicial complex with a piecewise at and(ϑ, h)-small metric, α∈H^1,1Ω^k. Then there areα⁰, α¹∈P⁻¹Ω^k and, ifλⁱ_s>0 for all components of theλ_s dening subdivision neighbourhoodsU(s), there isα²∈P⁻¹Ω^k, such that the L² norms ofα⁰,dα¹ and δα² are estimated by the corresponding norms of αup to a constant, and (with the Poincaré constantC˜

from 2.11b) hhhα−α⁰, βiii.C˜

h α ∇β for allβ∈H¹Ω^k, hhhdα−dα¹, βiii.C˜

h dα ∇β for allβ∈H¹Ω^k+1, hhhδα−δα², βiii.C˜

h δα ∇β for allβ∈H¹Ω^k−1. Proof. Let us introduce a space P⁻¹_elwΩ^k of elementwise P⁻¹ forms, spanned by

ω^s,e:=

®dvol_rs in U(s)∩re

0 elsewhere fors⊂e,e∈K^k. Then we can ndα˜⁰,α˜¹ andα˜²such that, for eache∈Kⁿ,

ˆ

re

hα˜⁰,βi¯ = ˆ

re

hα,βi¯ for all constantβ¯∈Ω^k, ˆ

re

hdα˜¹,βi¯ = ˆ

re

hdα,βi¯ for all constantβ¯∈Ω^k+1

(9.19a)

and similarly for δα². In fact, writing α˜⁰= ˜α⁰_t,eω^t,e and inserting β =v^[_I from above with I ∈ ⁿ_k

, the equations for determining coecients α˜⁰_t,e have M_(k) as system matrix, which has full rank by 9.18. The integral ´

hd˜α¹,βi¯ reduces to a boundary integral by 9.17 which does not includedanymore, this boundary integral is invariant under ane transformations, and the problem is solvable on the unit simplexD. If all λⁱ_sare positive, theδ^t_sare just a row- and column-rescaling ofd^s_t by non-zero factors.

As a second step, letα⁰=α⁰_tω^t ∈P⁻¹Ω^k be dened by the condition that, for each t∈K^k and alls∈K^k+1,

ˆ

U(t)

hα⁰,βi¯ = ˆ

U(t)

h˜α⁰,β¯i for all constantβ¯⁰ ∈Ω^k, ˆ

U(s)

hdα¹,βi¯ = ˆ

U(s)

hd˜α¹,β¯i for all constantβ¯⁰ ∈Ω^k+1

(and similarly forδα²), which means averagingα˜⁰over all parts U(t)∩rewitht⊂e. These forms have the desired properties: Norm-preservation is clear by construction.

For the approximation, observe that β ∈Ω^k can be replaced by some β¯ that is con

stant in each elementre, and the error is estimated by the Poincaré inequality 2.10c:

β−β¯_L2 ≤C˜

h ∇β L². And similarly it can be replaced by some β¯⁰ that is constant in each neighbourhoodU(t). This gives

hhhα−α⁰, βiii=hhhα−α˜⁰, βiii+hhh˜α⁰−α⁰, βiii

.hhhα−α˜⁰,β¯iii+hhh˜α⁰−α⁰,β¯⁰iii+ ˜Ch α ∇β ,

9. A Piecewise Constant Interpolation of dec and the scalar products vanish by choice ofα˜⁰andα⁰. Exactly the same computation is feasible forhhhdα−dα¹, βiiiandhhhδα−δα², βiii, q. e. d.

9.20 Proposition. For a complex consisting of only one n-simplex e, let d_(k)(e) be the

n+1 k+2

× ⁿ⁺¹_k+1

matrix representation(d^t_s)_s∈Kk+1,t∈K^k of d:P⁻¹Ω^k→P⁻¹Ω^k+1. Suppose K is a simplicial complex with a piecewise at and (ϑ, h)-small metric.

Assume that the ⁿ⁺¹_k+1

× ⁿ⁺¹_k+1-matrix Ç M(k)(e)

M_(k+1)(e)d_(k)(e) å

has full rank for eache∈Kⁿ. (9.20a) LetC˜be the Poincaré constant from 2.11b. Then for eachα∈Ω^k, there isα¯∈P⁻¹Ω^k with α¯L^p . αL^p, d¯αL^p. dαL^p and

hhhα−α, βiii¯ .C˜

h α ∇β for allβ ∈H¹Ω^k hhhdα−d¯α, βiii.C˜

h dα ∇β for allβ ∈H¹Ω^k+1. (9.20b) Proof. The assumption 9.20a guarantees that we can nd one singleα˜ ∈P⁻¹_elwΩ^k ful

lling both equations in 9.19a at the same time, q. e. d.

9.21 Remark. <a> We did not succeed to verify 9.20a in the general case, but there are at least no structural obstructions for it to hold:M_(k)has full rank ⁿ_k, andd_(k), which is the transposed of∂k+1 from 9.3 with rows scaled by|t| and columns scaled by |s|, has rank _k+1ⁿ , which add up to ⁿ⁺¹_k+1.

<b> Using anL^p Poincaré inequality (Evans and Gariepy 1992, thm. 4.5.2) instead of 2.10c leads tohhhα−α⁰, βiii ≤C˜,ph αL^p ∇β L^pand similar fordα−dα¹andδα−δα².

<c> The estimates 9.20b are formulated asH⁻¹norm estimates. By inserting mollied characteristic forms or forms with small support and |´

β| = 1 (Dirac forms), one can localise the convergence.

<d> We do not call 9.19 and 9.20 interpolation estimates, asα⁰, α¹, α² or α¯ may have nothing to do withαpointwise, but only in integral mean. In contrast to interpo

lation, the integral ofαandα⁰ over smaller spaces like boundaries of the U(t)will in general not converge, as 9.17 shows. For an example, see also 10.30. The section title interpolation of dec does not refer to interpolation of smooth functions, but to the process of extending the simplicial denitions in 9.4 toL^∞ forms in 9.11.

C. Applications

Im Dokument numerical approximation in riemannian manifolds by karcher means (Seite 65-77)