Real-Valued Variational Problems

C. Applications

respect to the trial spaceP¹₀ is the solutionu_h toLap(u_h, v) =hhhf, viii_{M g} for allv∈P¹₀. Naturally, there is also the notion of ag^e-Galerkin solution.

10.5 Remark. By 2.12, we know that the Dirichlet problem has no solution for general f ∈ L², but only forf ⊥H, and the solution is unique up to harmonic components, in other words: there is a unique solution inH^⊥. But the space of harmonic functions is one-dimensional, consisting only of the constant functionsand these are ruled out by the boundary value requirements.

10.6 Fact (Schwarz 1995, also cf. 2.19). The de Rham complex (H^1,0Ω, d) of a smooth compact Riemannian manifold is a Fredholm complex, so the Dirichlet problem is uniquely solvable, and du_L2 ≤C

f _L2 with the Poincaré constant C

from 2.10b.

This means thatL⁻¹ is a bounded linear operator.

If∂M is piecewise smooth or convex (that means, convex where it is not smooth), then M is H²-regular, i. e. there is a constant C depending on M, but not on f, with u_H2 ≤C f _L2, that means that L⁻¹ _L2,H² ≤C in this case.

10.7 Lemma (Céa). Situation as in 10.1. Let ube the Dirichlet potential and uh be the g-Galerkin solution tof ∈L². Then uh is the orthogonal projection of uontoP¹₀ with respect to Lap(·,·).

Proof. AsP¹₀⊂H¹₀, alsoufullls Lap(u, v) =hhhf, viii_L2 for allv ∈P¹₀ by whichuh was dened. So we have the so-called Galerkin orthogonality Lap(u−uh, v) = 0for all suchv∈P¹₀, which is the characterising property of the projection error, q. e. d.

10.8 Corollary. Situation as in 10.1. Let Π be the orthogonal projection H¹₀ → P¹₀ with respect to Lap(·,·) and Π^⊥ := id−Π be the projection error. Then for any k for which both sides are dened,

u−u_hHk ≤ f _L2 Π^⊥L⁻¹ _L2,H^k.

10.9 Proposition. Situation as in 10.1 with dimensionn≤3. Then Π^⊥ _H2,H¹ .ϑ⁻¹h, and if additionallyM isH²-regular, then

u−u_{h H}1.C

hϑ⁻¹ f _L2.

Proof. It suces to show that there is oneu_h∈P¹with u−u_hH1 .ϑ⁻¹h u_H2, then the projection ofuwill produce a smaller error than thisu_h. As usual, we takeu_hto be the Lagrange interpolation ofu(which is well-dened, asH²⊂C⁰in dimension≤3, cf.

Adams 1975, Theorem 5.4.c). And this interpolation estimate is exactly 7.5, q. e. d.

10.10 Proposition (AubinNitsche). Situation as in 10.1. Then Π^⊥L⁻¹ _L2,L² ≤ Π^⊥L^{−1 2}_L2,H¹. Under the same conditions as in 10.9,

u−uhL² .C²

h²ϑ⁻² f L².

10. Real-Valued Variational Problems Proof. First, note that for a right-hand sideg, the solutionL⁻¹g is characterised by hhhg, viii_{M g} = Lap(L⁻¹g, v)for allv∈H¹₀. Now for a right-hand sidef ∈L², consider

Π^⊥L⁻¹f L²(M g)= sup

g∈L²

hhhΠ^⊥L⁻¹f, giii_L2

g_L2

= supLap(Π^⊥L⁻¹f, L⁻¹g) g_L2

(∗)= supLap(Π^⊥L⁻¹f,Π^⊥L⁻¹g) g _L2

≤ Π^⊥L⁻¹f _H1 Π^⊥L⁻¹ _L2,H¹, where we have used in(∗)thatΠand henceΠ^⊥is aLap-orthogonal projection, q. e. d.

10.11 Remark. It would of course be possible to consider other interpolation procedures

than just nodal Lagrange interpolation, for example averaged Taylor polynomials as in Brenner and Scott (2002, section 4.1), which would circumvent the dimension restrictions. However, the emphasis of this thesis lies more on the dierent possible ap

plications of the Karcher simplex construction than on optimal results for the Dirichlet problem.

10.12 Lemma. Situation as in 10.1. LetF(v) :=hhhv, fiii_M,g, and letLap^eandF^ebe dened

similar toLapandF, but withg^einstead ofgeverywhere. Then|(Lap−Lap^e) (v, w)|. C₀⁰h² dv_L2 dw_L2 and|(F−F^e)v| .C₀⁰h² v _L2.

Proof. Exactly as in 7.3, q. e. d.

Remark. In the understanding of Hildebrandt et al. (2006), the weak LaplacianLg is a mappingH¹→(H¹)^∗,Lgu:v7→Lap(u, v). In this setting, 10.12 can be seen as a convergence result for the weak Laplacians: Lg−Lg^e H¹,(H¹)^∗.C₀⁰h².

10.13 Proposition. Situation as in 10.1 withH²-regularM. Letuh, u^e_h∈P¹₀be the Galerkin

solutions toLgu=F andLg^eu^e=F^e. Then

uh−u^e_hL2+C duh−du^e_{h L}2.C₀⁰C²

h² f _L2.

Proof. During this proof, · always means · _L2(M g). Let us rst consider the deriva

tive term on the left-hand side: For somev with v = 1, we have du_h−du^e_h = Lap(u_h−u^e_h, v)

≤ |Lap(uh, v)−Lap^e(u^e_h, v)|+|Lap^e(u^e_h, v)−Lap(u^e_h, v)|

≤ |(F−F^e)v|+|(Lap^e−Lap)(u^e_h, v)|

.C₀⁰h² f v +C₀⁰h² du^e_h dv Then use du^e_h ≤C

f from 10.6. For the estimate of uh−u^e_h, use the Poincaré

inequality again, q. e. d.

Remark. As in the euclidean setting, the proofs carry over to an arbitrary continuous, stronglyH¹₀-elliptic bilinear form onH¹ instead ofLap.

Variational Problems in Ω^k

10.14 Assumption. Situation as in 10.1. For k= 0, . . . , n, let there be nite-dimensional subspacesPΩ^k ofH^1,0Ω^k(orH^0,1Ω^k, if needed) withL²andH^1,1approximation order hanalogous to 7.4:

min

v_h∈PΩ^k v−vhL²+ dv−dvhL²+ δv−δvhL²

| {z }

only for 10.15a

≤αh vH²

and similar for

t

^{∗v =}

t

^∗vh = 0 or

n

^{v =}

n

^vh = 0. Furthermore, assume that the Dirichlet problem is H²-regular and the Hodge decomposition u = da+δb+c is H¹-regular, which means da_H1 . u_H1 etc. We abbreviate hhh ·,· iiiL²(M g^e)as hhh ·, · iiie. 10.15 Proposition. Assume 10.14. Let u = da+δb+c be the Hodge decomposition of

u∈ H^1,1Ω^k, which can be computed as a= argminF[u] overa ∈H^1,1Ω^k−1_t and b = argminG[u] overb∈H^1,1Ω^k+1_n as in 2.14. If a_h= argminF[u] overa_h∈PΩ^k−1_t and b_h= argminG[u]overb_h∈PΩ^k+1_n , then

da−dah L²+ δb−δbhL² ≤αh u_H1. (10.15a) If u=dae+δbe+ce is the Hodge decomposition with respect to g^e, and if ah,e and bh,e are dened similiarly, then

da−daeL²+ δb−δbeL²+ c−ceL² .C₀⁰h² uL², (10.15b) dah−dah,e L²+ δbh−δbh,eL² .C₀⁰h² u_L2. (10.15c) Proof. ad primum: By the EulerLagrange equation hhhda, dviii = hhhu, dviii for all v ∈ H^1,1Ω^k+1_t andhhhdah, dviii=hhhu, dviiifor allv ∈PΩ^k+1_t , we know that dah is the L²-best approximation ofdain d(PΩ^k_t), which is smaller thanαh ∇da by assumption.

ad sec.: If hhhdah,e, dviiie− hhhu, dviiie = 0, then hhhdah,e, dviii − hhhu, dviii.C₀⁰h²( da_h,e + u) v and hencehhhdah−da_h,e, dviii.C₀⁰h²( da_h,e + u) v for allv∈PΩ^k_t. The same calculation is valid forda_h−da_h,e instead ofda−da_e. Thec−c_eestimate comes out

as the remainder, q. e. d.

10.16 Remark. <a> 10.15b is our analogue of thm. 3.4.6 in Wardetzky (2006).

<b> In general, there will be no exact nite-dimensional Hodge decomposition inPΩ^k, as we have not required any connection betweend(PΩ^k)andPΩ^k+1. There is a Hodge decomposition in the space of Whitney forms with convergence proven by Dodziuk (1976, thm. 4.9). Variational problems in a specic spaceP⁻¹Ω^k of piecewise constant forms will be treated in 10.24sqq.

<c> The feec setting of Arnold et al. only has a weak Hodge decompositionu = dah+ ˜bh+ch ofu∈PΩ^k as in 2.13, but as its parts are also orthogonal projections, there is an estimate

dah−dah,e L²+ ˜bh−˜bh,eL²+ ch−ch,e L² .C₀⁰h² u_L2

corresponding to 10.15b.

10. Real-Valued Variational Problems

10.17 Mixed form of Dirichlet problem. Arnold et al. (2006, 2010) have shown how

to construct nite-dimensional subcomplexes(PΩ, d)of(H^1,0Ω, d)and solve the mixed Dirichlet problem therein. Holst and Stern (2012) have extended this to the situation where the domain of the Sobolev space and the nite-dimensional approximation are endowed with dierent, but close Riemannian metricsg and g^e, which leads to the situation that the inclusion map PΩ^k(M g^e) → H^1,0Ω^k(M g) is not norm-preserving anymore, but only an almost-isometric map. Their setting directly applies to Finite Element computations on the KarcherDelaunay triangulation:

Proposition (Holst and Stern 2012, thm. 3.10). Assume 10.14, and use the no

tation from 2.17sq. For f ∈ L²Ω^k, let (σ, u, p) ∈ PS and (σe, ue, pe) ∈ PSe be the solution of the mixed formulation 2.18 of the Dirichlet problem in M g and M g^e re

spectively, where PS = PΩ^k_t ×PΩ^k+1_t ×PH^k_t is a stable choice of trial spaces from Arnold et al. (2006, eqn 7.14), and PSe diers from PS only by the last factor (PH^k_e)_t, the harmonic trial functions with respect tog^e. Then

σ−σeH¹+ u−ueH¹+ p−peL² .C₀⁰

γ h² f _L2, whereγ is the inf-sup constant as in 2.18 (but overPΩ^k).

Proof. The solutions= (σ, u, p)with respect to the correct scalar productg fullls b(s, t) =F(t)for every test triplet= (τ, v, q)∈PS. On the other hand, the distorted solutionse fulllsbe(se, te) =Fe(te)for allte∈PSe with the obvious denition ofbe

andFe. As the trial spaces only dier in the last termq, we have be(se, te) =be(se, t) +hhhue, qe−qiiie=be(se, t) +hhhue, qiiie

(becauseu_e ⊥PH^k_e). Now observehhhu_e, qiii=hhhπu_e, qiii, where π is the orthogonal pro

jection ontoPH^k, and by 10.16c the projection of a PH^k_e element onto PH^k is small.

Hence

hhhue, qiiie=hhhue, qiii+ (hhhue, qiiie− hhhue, qiii).C_0,1⁰ h² u q .

Weakening the right-hand side, we obtain|b_e(s_e, t)−F_e(t)|.C₀⁰h² s t. By the scalar product comparison 10.12, also|b(s_e, t)−F(t)|.C₀⁰h² s t, and taking this together withb(s, t) =F(t), we have

b(s−se, t).C₀⁰h² s t .

Now, by the inf-sup-condition 2.18b,γ s−se ≤sup_tb(s−se, t)/ t, q. e. d.

Dirichlet Problems with Curved Boundary

The case that the analytical and the computational domain actually coincide is not the only interesting problem. When for example a Dirichlet problem on the unit disk in hyperbolic space is considered, a Karcher triangulation with respect to the whole hyperbolic space will not exactly cover the unit disk. But the treatment of such a boundary approximation is standard in Finite Element theory, and the main task is

to carefully inspect which arguments have to be modied because they rely on the Euclidean structure of the domain. We give a presentation according to Dörfler and Rumpf (1998) and do not treat the dierence between g and g^e, as this comparison can be done separately by using 10.13 after 10.21.

The usual setup for boundary approximation is that a domain Ω is replaced by a simplicial domain Ωh whose boundary vertices lie on ∂Ω. By (n−1)-dimensional interpolation estimates, one then gets that ∂Ω and ∂Ωh are only . h²κ far apart, whereκbounds the curvature of∂Ωandhthe mesh size ofΩh. We translate this, for Ω⊂M, into the following

10.18 Situation. Let M =rK be a piecewise at and (ϑ, h)-small realised simplicial com

plex. LetΩ⊂M be a full-dimensional domain andΩh=rK¯a realised full-dimensional subcomplex, connected by a normal graph mapΦ :∂Ω→∂Ωh,p7→exp_pdν, where d:∂Ω→R is Lipschitz-continuous andν is the outer normal on∂Ω, with the follow

ing properties: First, the retraction inverse (p, t) 7→ exp_ttdν is injective (to ensure that no topology change may happen). Seond, it is short in the send that|d| ≤αh², Lip d≤αh≤1, and|∇dd| ≤α(wheredis smooth) for someα∈R. Let all principal curvatures of ∂Ωin M be bounded by κ. This implies that for small hthe norms of dΦand∇dΦare bounded, see section 11.

10.19 Lemma. Situation as in 10.18. If v ∈ H¹(Ωh), then v _L2(Ωh\Ω) . αh² dv_L2(Ωh\Ω)

and vL²(∂Ω∩Ωh).√

αh dvL²(Ω_h\Ω) for small h.

Proof. ad primum: It suces to show the claim for smoothv. Considerλ∈Ωh\Ω. As d(λ, ∂Ωh).αh², there is a curveγ[λ] :µ;λfor someµ∈∂Ωhwith length.αh². If his small, this curve can be supposed to be a straight line lying entirely in one simplex ofK¯. Asv(µ) = 0,

v(λ) = ˆ

γ[λ]

dvγ.˙ (10.19a)

Supposeγis arclength-parametrised. Now we can again apply the arguments from the proof of 7.5 (keeping in mind thatγ[λ]has length hthere, butαh² here):

ˆ

Ω_h∩Ω

|v|²

(10.19a)

≤ ˆ

Ω_h∩Ω

ˆ

γ[λ]

|dv|2 (7.5a)

. αh² ˆ

Ω_h∩Ω

ˆ

γ[λ]

|dv|²

(7.5b)

. α²h⁴ dv²_L2(Ωh\Ω)

ad sec.: Because∂Ωhis a graph over∂Ω, the inverse is also true:∂Ωis a graph (usually not normal) over ∂Ωh, so we can introduce coordinates in which a simplex fof ∂Ωh

lies in the xm-plane and ∂Ω is parametrised by (x1, . . . , x_m−1) 7→ (x1, . . . , xm+1, ρ). Then

v²_L2(∂Ω∩rf)= ˆ

t

|v|²»

1 +|dρ|²

(11.10b)

. ˆ

rf

|v|²

(10.19a)

. ˆ

rf

ˆ

γ[λ]

|dv|^{2 (7.5a)}

. αh² dv²_L2(Ω_h\Ω),

q. e. d.

10. Real-Valued Variational Problems

10.20 Lemma. Situation as in 10.18. For v : M → R, which is H² continuous in Ω and

M \Ω, let [v] be the jump of v across ∂Ω. If h is small, there is a continuous ex

tension u¯ of u∈ H²(Ω) ontoΩ∪Ω_h such that u|¯_Ω = u, u¯_H2(Ω_h\Ω) . u_H2(Ω) and [d¯u ν]_L2(∂Ω∩Ω_h). u_H1(Ω).

Proof. By assumption, all points inΩ_h\Ωare covered by the homotopy Φt: p7→exp_ptν,

where at eachp∈∂Ω∩Ωh, the parameter t is chosen within ]0;d(p)] (in particular, points with negatived(p)are excluded, as they would parametrise Ω\Ωh instead of Ω_h\Ω). For an image point of Φ_t, setu(exp¯ _ptν) :=u(exp_p−tν), the reection along

∂Ω. Thisu¯is continuous, and[d¯u ν] =±2du ν. TheH²norm-preservation follows from the assumptions on Φ(but note thatu¯ is notH² in Ω_h∪Ω due to the jump on∂Ω,

even thoughΦ_tis smooth), q. e. d.

10.21 Proposition. Situation as in 10.18. Let u∈ H²₀(Ω) be the solution of Lu =f with

respect toΩ, and let uh ∈P¹₀(Ωh) be the Galerkin solution over Ωh for an extension of the right-hand sidef by zero ontoΩh\Ω. Then du−duhL²(Ω).√

αh u_H2(Ω)for smallh, whereuh has been extended by zero inΩ\Ωh.

Proof. Letu¯ be the extension ofufrom 10.20. Assume we can show

¯

u−u_H1(Ω_h). u¯−v_H1(Ω_h)+αh² u¯_H2(Ω_h\Ω)+√

αh [d¯u(ν)]_L2(∂Ω∩Ω_h) (10.21a) for every v ∈P¹(Ωh). Then the claim is proven by 7.5 and 10.20. Supposed v ∈P¹, observe that in dv−duh = suphhhdv−duh, dwiii/ dw , it suces to take w∈P¹. So we have

d¯u−duhL²(Ωh)≤ d¯u−dv + dv−duh = d¯u−dv + sup

w∈P¹

hhhdv−duh, dwiii dw

= d¯u−dv + sup

w∈P¹

hhhdv−d¯u, dwiii+hhhd¯u−duh, dwiii dw

≤2 d¯u−dv + sup

w∈P¹

hhhd¯u−du_h, dwiii

dw .

And now, iff¯is the extension off byf¯= 0inΩh\Ω, hhhd¯u−duh, dwiii_L2(Ω_h)=

ˆ

Ω_h∩Ω

hd¯u, dwi −f w+ ˆ

Ω_h\Ω

hd¯u, dwi −f w¯

= ˆ

Ω_h∪Ω

(−∆u−f)

| {z }

=0

w+ ˆ

∂Ω_h∩Ω

w du ν+ ˆ

Ωh\Ω

−w∆u+ ˆ

∂(Ωh\Ω)

w d¯u(−ν),

as−ν is the outer normal ofΩh\Ω. So this gives

hhhd¯u−duh, dwiii_L2(Ω_h)≤ ∆¯uL²(Ω_h\Ω) wL²(Ω_h\Ω)+ [du(ν)]L²(∂Ω∩Ωh) wL²(∂Ω∩Ωh), which shows, together with 10.19 for the w norms, the claimed estimate 10.21a,

q. e. d.

Heat Flow

Goal. As a short outlook on Galerkin methods for parabolic problems, we consider the approximation of heat ow under perturbations of metric. We decided to exclude the general convergence theory (see e. g. Thomée 2006, chap. 1) and concentrate on the dierence between Galerkin approximations with respect tog andg^e.

10.22 Proposition. Situation as in 10.1. For a time interval[0;a], let u_h, u_h,e be the time-continuous Galerkin approximation to the heat ow with initial value u₀ ∈ P¹₀ and right-hand side f ∈L^∞([0;a],L²(M g))for metricsg andg^erespectively, that means

hhhu˙h, viii+hhhduh, dviii=hhhf, viii for allv∈P¹, uh|t=0=u0, hhhu˙_h,e, viii_e+hhhduh,e, dviii_e=hhhf, viii_e for allv∈P¹, u_h,e|t=0=u₀, wherehhh ·,· iii_e is the abbreviation forhhh ·,· iii_{M g}e. Then their dierence can be estimated by

uh−uh,eL^∞(L²).C₀⁰Ch² u0H¹+C₀⁰Ch²ˆ

f(t)²_L2

^1/2 .

Proof. The proof follows the line of the usual convergence proof for parabolic problems as in Thomée (2006, thm. 1.2): Considerε:=uh−uh,e. By the dening equations for uh anduh,e, we have

hhhε, viii˙ +hhhdε, dviii=hhhf, viii − hhhu˙h,e, viii − hhhduh,e, dviii.

By 7.3 and 10.12, we have

|hhhu˙h,e, viii − hhhduh,e, dviii − hhhf, viii|.C_0,1⁰ h² u˙h,e v + duh,e dv + f v , where all norms are L² norms. So we have for v = ε, together with the Poincaré constantC from 2.10c,

1 2

d

dt ε²+ dε².C₀⁰Ch² u˙h,e + duh,e + f dε .

Then Young's inequality gives2cab≤c²a²+b², hence we obtain a separated summand dε²on the right-hand side, which can be cancelled (the suppressed constant belongs toc):

1 2

d

dt ε².(C₀⁰C

h²)² u˙h,e2+ duh,e2+ f ² Integration over[0;a]gives, asε|t=0= 0,

ε²≤(C₀⁰C

h²)² ˆ

˙

uh,e2+ duh,e2+ f ².

From the usual regularity theory for parabolic problems (Thomée 2006, eqn. 1.20, casem= 0), we know that´

( ˙u²+ u²_H1). u0H¹+´

f ², which shows the desired

estimate forε, q. e. d.

10. Real-Valued Variational Problems

10.23 Proposition. Situation as above. Let uⁿ_h, uⁿ_h,e be the Galerkin approximation to the

heat ow with implicit Euler time discretisation with respect tog and g^e respectively, that means

hhh∂u¯ ⁿ_h, viii+hhhduⁿ_h, dviii=hhhf, viii for allv∈P¹₀, u⁰_h=u0, hhh∂u¯ ⁿ_h,e, viii

e+hhhduⁿ_h,e, dviii

e=hhhf, viii_e for allv∈P¹₀, u⁰_h,e=u0

for the backward dierence quotient∂v¯ ⁿ :=_τ¹(vⁿ−vⁿ⁻¹). Then their dierence at time t=nτ can be estimated by duh−duⁿ_{h,e L}2.Kh²t, whereKdepends on the geometry,

f _L^∞_(L2)and u0H¹.

Proof. As before, letεⁿ:=uⁿ_h−uⁿ_h,e. Then

hhh∂ε¯ ⁿ, viii+hhhdε, dviii=hhhf, viii − hhh∂u¯ ⁿ_h,e, viii − hhhduh,e, dviii, and the right-hand side is bounded by

|hhh∂u¯ ⁿ_h,e, viii − hhh∂u¯ ⁿ_h,e, viii_e|+|hhhduh,e, dviii − hhhduh,e, dviii_e|+|hhhf, viii − hhhf, viii_e| .C₀⁰h² ∂u¯ ⁿ_h,e v + duh,e dv + f v .C₀⁰C

h² ∂u¯ ⁿ_h,e + du_h,e + f dv .

Denote the whole term in parentheses asΛ. As before, it is bounded in terms of the given data. Then again the choicev=εⁿ gives

ε^{n 2}− hhhεⁿ⁻¹, εⁿiii+ dεⁿ².C₀⁰C

h²τΛ dε^{n 2} and so

εⁿ²+ dε^{n 2}.C₀⁰C

h²τΛ dεⁿ²+C²

dεⁿ⁻¹ dεⁿ .

And of course dε^{n 2} is smaller than the last left-hand side, which gives dεⁿ . C_0,1⁰ C

h²τΛ +C²

dεⁿ⁻¹ . Then the claim follows by induction overn, q. e. d.

Discrete Exterior Calculus

10.24 Observation. As we have noticed in 2.19, all variational problems from section 2

are uniquely solvable in (P⁻¹Ω^k, d) like in (Ω^k, d)by the construction of P⁻¹Ω^k as a (co-)chain complex. As(P⁻¹Ω^k, d)just a gentle way of writing the simplicial cochain complex (C^k, ∂^∗), its (co-)homology is isomorphic to the de Rham complex' one (a short direct proof, called the theorem of de Rham, is given in Whitney 1957, sec.

iv.29, although de Rham 1931 proved isomorphy to singular, not simplicial coho

mology). Therefore, we can hope for approximating smooth solutions of variational problems by ones inP⁻¹Ω^k.

10.25 Situation. Let rK be a realised oriented regular n-dimensional simplicial complex

without boundary with a piecewise at,(ϑ, h)-small metricg. Let λs,s∈K^∗, be the simplices' circumcentres, and suppose λⁱ_s > 0 for all their components (i. e. rKg is well-centred). Assume 9.20a and that the Hodge decomposition u = da+δb+c is H¹-regular, meaning daH¹ . uH¹ etc. We use the Poincaré inequality in the form 2.11b.

10.26 Proposition. Situation as in 10.25. For a functionf ∈H¹, letu∈H²be the solution of the Poisson problem hhhdu, dviii =hhhf, viii for all v ∈ H¹, and let u_h ∈ P⁻¹Ω⁰ be the solution ofhhhdu_h, dv_hiii=hhhf, v_hiiifor all v_h∈P⁻¹Ω⁰. Then

hhhdu−du_h, dv_hiii.C˜

h( ∇f vh + ∇du dvh ) for all v_h∈P⁻¹Ω^k. Proof. Letvandvhbe connected by 9.20b. Then

hhhdu−duh, dvhiii=hhhdu, dviii − hhhduh, dvhiii+hhhdu, dvh−dviii=hhhf, v−vhiii+hhhdu, dvh−dviii,

and both terms can be estimated as claimed, q. e. d.

10.27 Proposition. Situation as in 10.25. Letu=da+δb+cbe the Hodge decomposition of u∈H^1,1Ω^k, and letu¯=dah+δbh+ch be the Hodge decomposition of itsL²-orthogonal projection ontoP⁻¹Ω^k. Then

hhhda−dah, dvhiii.C˜

h u_H1 dvhL²

hhhδb−δb_h, δv_hiii.C˜

h u_H1 δv_hL2

for allv_h∈P⁻¹Ω^k.

Proof. We know that da is characterised by hhhda, dviii = hhhu, dviii for all v ∈ H^1,0Ω^k. Naturally, ah is characterised by hhhdah, dvhiii = hhh¯u, dvhiii for all vh ∈ P⁻¹Ω^k, but the right-hand side is hhhu, dvhiii if du is the orthogonal projection onto P⁻¹. So we can proceed exactly like before, but using 9.19 to connect onlydvanddvhinstead ofvand vh:

hhhda−dah, dvhiii=hhhda, dviii−hhhdah, dvhiii+hhhda, dvh−dviii=hhhu, dv−dvhiii+hhhda, dvh−dviii, the ∇da produced by the latter term can be estimated by u_H1 by assumption. The same procedure is feasible forδbandδbh (where another test formv can be employed

such thatδv is close toδvh), q. e. d.

10.28 Proposition. DeneS¹:=H¹Ω^k−1×H¹Ω^k×H¹H^k andP⁻¹S:=P⁻¹Ω^k−1×P⁻¹Ω^k× P⁻¹H^k. Suppose s = (σ, u, p) ∈ S¹ is a solution of the Poisson problem in mixed form as in 2.17, and sh = (σh, uh, ph) ∈ P⁻¹S is the solution of the corresponding nite-dimensional problem. Then for allth= (τh, vh, qh)∈P⁻¹S,

b(s−sh, th).C˜h( ∇f _L2+ ∇s_L2) thH^1,0,

where the left-hand side is of course not to be taken literally as in 2.18a, but with P⁻¹ exterior derivatives forsh andth, i. e. consisting of terms like hhhdu−duh, dvhiiietc.

Proof. In the spirit of 10.26, we start with

b(s−s_h, t_h) =b(s, t)−b(s_h, t_h) +b(s, t−t_h).

As before, the rst two terms arehhhf, v−vhiii, which is well-controlled by the right-hand side of the claim. Inb(s, t−th), there are many easy terms, which we do not explicitely discuss once more. Only hhhu, q −q_hiii = hhhu, qhiii is iteresting. Estimating it actually

11. Approximation of Submanifolds

Im Dokument numerical approximation in riemannian manifolds by karcher means (Seite 77-87)