Analysis of the Stationary Problem

We first examine the stationary variant ofeqs. (6.1), i.e. the problem of finding cε,f: Ωf →Randcε,s: Ωs→Rsuch that

−∆cε,f +¹_εvf · ∇cε,f = 0 inΩf, (6.2a)

−∆c_ε,s+c_ε,s= 0 inΩ_s, (6.2b) with exterior boundary conditions

c_ε,f =cⁱⁿ_f on∂iΩ_f, (6.2c)

∂ncε,f = 0 on∂oΩf∪∂wΩf, (6.2d)

∂_ncε,s= 0 on∂Ωs\Γ (6.2e) and interface coupling conditions

∂n_fcε,f+ξ(cε,f−cε,s) = 0 onΓ, (6.2f)

∂_nsc_ε,s+ξ(c_ε,s−c_ε,f) = 0 onΓ. (6.2g) where ξ ∈ L^∞(Γ) with ξ > 0 a.e. and vf ∈ L^∞(Ωf) with divvf = 0. The regularity ofcⁱⁿ_f and further assumptions onvf are expressed below in terms of regularity of solutions to the pure advection problem. For the weak formulation ofeqs. (6.2)we define the spaces

Cf :={ϕf ∈H¹(Ωf)|ϕf|_∂iΩf = 0}, ¯Cf :=H¹(Ωf), Cs:=H¹(Ωs).

and, reminding of the notation (·,·)_i for the scalar product on L²(Ω_i) for i∈ {f, s}, the bilinear forms

aε,f(cf, ϕf) := (∇cf,∇ϕf)f+¹_ε(vf· ∇cf, ϕf)f ∀cf ∈¯Cf, ϕf ∈¯Cf, as(cs, ϕs) := (∇cs,∇ϕs)s+ (cs, ϕs)s ∀cs, ϕs∈Cs.

In practise only the restriction ofaε,f toCf×Cf will be of interest, requiring the typical reduction of the problem to homogeneous boundary values. We equipL²(Γ)with theξ-weighted scalar product

(·,·)ξ := (ξ¹²·, ξ¹²·)Γ, k · kξ := (·,·)

1 2

ξ.

With these definitions,eqs. (6.2)can be rewritten in weak form as:

Problem 6.2.1. Find cε,f ∈¯Cf andcε,s∈Cs such that

a_ε,f(c_ε,f, ϕ_f) + (c_ε,f−c_ε,s, ϕ_f)_ξ = 0 ∀ϕ_f ∈Cf, (6.3a) as(cε,s, ϕs) + (cε,s−cε,f, ϕs)ξ = 0 ∀ϕs∈Cs (6.3b) with c_ε,f =cⁱⁿ_f on ∂iΩ_f.

That strong solutions toeqs. (6.2)are weak solutions in the sense of Prob-lem 6.2.1and vice versa (under additional regularity assumptions) can be easily seen and uses that the interface condition is satisfied due to the boundary terms occurring while partially integrating ∆cε,f onΩf and∆cε,sonΩs.

We make the following assumption on the regularity of the pure advection problem.

Assumption 6.2.2. There is exactly onec0,f ∈¯Cf such that (vf· ∇c0,f, ϕf)f = 0 ∀ϕf ∈L²(Ωf)

withc_0,f|_∂iΩf =cⁱⁿ_f.

The notation c_0,f anticipates our later proof that c_0,f is the zeroth order approximation of c_ε,f as ε → 0, see Theorem 6.2.11. Assumption 6.2.2 is restrictive in the sense that c0,f ∈ H¹(Ωf) requires regularity perpendicular to the advection velocity, a requirement which is not easily satisfied given thatc0,f is determined by the value ofcⁱⁿ_f along the characteristics ofvf. This assumption is nevertheless central for the subsequent analysis, not least because a trace ofc0,f is required for the well-posedness of the presented limit equation forcε,s. In the model by Yang et. al. there holdscⁱⁿ_f ≡1andAssumption 6.2.2is trivially satisfied withc0,f ≡1. This even greater regularity ofc0,f is exploited in some of the results below.

Existence and uniqueness of solutions for a similar, instationary model was proven in [QVZ02b], but the arguments are repeated here to show the (in-) dependence of the constants onε.

Lemma 6.2.3. For everyε >0 Problem 6.2.1has a unique solution with kc_ε,fkH¹(Ω_f)+kc_ε,skH¹(Ω_s)≲1 (6.4) where the constant depends on the data, e.g. kc0,fkH¹(Ω_f), but not onε.

Proof. Define the bilinear form

aε((cf, cs),(ϕf, ϕs)) :=aε,f(cf, ϕf) +as(cs, ϕs) + (cf−cs, ϕf−ϕs)ξ

forcf, ϕf ∈¯Cf andcs, ϕs∈Cs. Then the solution ofProblem 6.2.1satisfies a_ε((c_ε,f−c_0,f, c_ε,s),(ϕ_f, ϕ_s)) =−(∇c_0,f,∇ϕ_f)_f −(c_0,f, ϕ_f−ϕ_s)_ξ (6.5) for all(ϕf, ϕs)∈Cf×Cs. We claim thataεis coercive on(Cf×Cs)×(Cf×Cs) with constant independent ofε. This follows byξ >0 and using that

Z

Ω_f

vf· ∇ϕfϕfdx= Z

Ω_f

div(vfϕf)ϕfdx

=− Z

Ω_f

(v_fϕ_f)· ∇ϕ_fdx+ Z

∂Ω_f

v_f·nf|ϕ_f|²do forϕf ∈Cf, which implies

Z

Ω_f

vf· ∇ϕfϕfdx=1 2

Z

∂Ω_f

vf·nf|ϕf|²do≥0

using for the last inequality thatvf·nf≥0on∂Ωf\∂iΩf andϕf= 0on∂iΩf. Since (cε,f −c0,f, cε,s)∈Cf×Cs the existence and uniqueness of solutions to eq. (6.5), and thus toProblem 6.2.1, is guaranteed by the Riesz representation theorem. Inequality (6.4) follows by coercivity of aε and that the right-hand

side ofeq. (6.5)is independent ofε. □

Remark 6.2.4. We emphasize that theε-independence inineq. (6.4)depends fundamentally on Assumption 6.2.2. For an arbitrary extension of cⁱⁿ_f in H¹(Ωf), as typically employed for non-homogeneous boundary values, the right-hand side of eq. (6.5) in the proof of Lemma 6.2.3 would contain an

advection term of orderO(ε⁻¹).

In the limit,c_ε,fandc_ε,swill converge in some appropriate sense to solutions of (repeating the definition ofc_0,f for convenience):

Problem 6.2.5 (Stationary limit). Find c_0,f ∈¯Cf andc_0,s∈Cssuch that (vf· ∇c0,f, ϕf)f = 0 ∀ϕf ∈L²(Ωf),

a_s(c_0,s, ϕ_s) + (c_0,s−c_0,f, ϕ_s)_ξ = 0 ∀ϕ_s∈Cs

with c0,f =cⁱⁿ_f on∂iΩf.

Lemma 6.2.6. The solutionc0,stoProblem 6.2.5exists, is unique and satisfies kc0,skH¹(Ω_s)≲1

with constant depending on the data. The existence of c_0,f is guaranteed by Assumption 6.2.2

Proof. Moving c_0,f in the equation for c_0,s to the right-hand side, the left-hand side is a coercive, continuous and symmetric bilinear form, hence exis-tence, uniqueness and the claimed estimate follows by the Riesz representation

theorem. □

The estimate fromineq. (6.4)is sufficient to prove qualitative results about the convergence of cε,f andcε,s asε→0without any further assumptions.

Lemma 6.2.7. Asε→0 there holds

c_ε,f * c_0,f inH¹(Ω_f), cε,s→c0,s inH¹(Ωs).

Proof. The estimate fromineq. (6.4)implies that there arec^∗_0,f ∈H¹(Ωf)and c^∗_0,s∈H¹(Ωs)such that for subsequences ofcε,f andcε,s there holds

cε,f * c^∗_0,f inH¹(Ωf), c_ε,s* c^∗_0,s in H¹(Ω_s)

asε→0. We do not distinguish between sequence and subsequence in notation here. To see that c^∗_0,f =c0,f we multiplyeq. (6.3a)byεand take the limit as ε→0. The claim follows for this subsequence using the weak convergence and the ε-independent bounds from Lemma 6.2.3. Due to the uniqueness of the limit the claim then follows for the whole sequencecε,f, an argument which is also implicitly used forcε,s in the following.

Ineq.(6.3b)we directly pass to the limit, establishingc^∗_0,s=c0,s. For strong convergence ofcε,s we use the standard Sobolev embedding

H¹²(Γ),−,→L²(Γ)

which implies cε,f →c0,f inL²(Γ). The erroreε,s:=cε,s−c0,s satisfies a_s(e_ε,s, ϕ_s) + (e_ε,s, ϕ_s)_ξ = (c_ε,f−c_0,f, ϕ_s)_ξ

forϕs∈Csand testing withϕs=eε,s yields

keε,sk²s+k∇eε,sk²s≲kcε,f−c0,fkξ →0 (6.6)

which concludes the proof. □

We use the following additional assumption on the advection velocity field, following [Eck79, Chapter 6], to establish quantitative convergence results.

Assumption 6.2.8. There existsψ∈C²(Ωf)such that

−v_f· ∇ψ≥ν >0

whereν ∈R.

Assumption 6.2.8implies that any trajectory of the pure advection problem leaves Ω after finite time, in fact Assumption 6.2.8holds if vf has no closed curves andvf 6= 0inΩf [DEF74]. The required regularity ofψcan be reduced to ψ∈W^2,∞(Ωf), for which a similar characterization was proven in [AM09].

Note that the condition ν >0 cannot be satisfied ifvf = 0 along the bound-ary of the artery, preventing the application to the plaque problem. We will nevertheless treat this, simpler, case first to demonstrate the main techniques, before extending the analysis under more permissive conditions.

Example 6.2.9. In [Eck79, Chapter 6] the functionψ(x) =α·xforα∈R^dwas investigated. In our numerical examples we considerΩf = (−^L₂,^L₂)×(0,1)with L >0andvf(x, y) = (1,0)^T, where the assumption is satisfied forα:= (−1,0)^T

andν := 1.

To handle the interface term we will employ the following interpolation-type trace estimate.

Lemma 6.2.10. For any cf ∈Cf there holds

kcfkL²(∂Ω_f)≲kcfkf¹²k∇cfkf¹². (6.7) Proof. According to [Gri11, Lemma 1.5.1.9] there exists δ > 0 and νf ∈ C^∞(Ωf) such that νf ·nf ≥ δ on ∂Ωf, making νf in this sense a smooth approximation ofnf. Then, following the proof of Theorem 1.5.1.10 in [Gri11], there holds by Green’s theorem that

Z

∂Ωf

c²_fνf·nfdo= Z

Ωf

∇(c²_f)·νfdx+ Z

Ωf

c²_fdiv(νf)dx

= 2 Z

Ωf

cf∇cf·νfdx+ Z

Ωf

c²_fdiv(νf)dx

Usingνf·nf ≥δ, bounding the terms involvingνf bykνfk_C1(Ωf)and applying Hölder’s inequality, it follows that

kc_fk²L²(∂Ω_f)≲kc_fkfk∇c_fkf +kc_fk²f.

The claimedineq. (6.7)is then a consequence of applying Poincaré’s inequality to one factor kcfkf of the last term on the right. □

Theorem 6.2.11. Let cε,f and cε,s solve Problem 6.2.1, c0,f and c0,s solve Problem 6.2.5. LetAssumption 6.2.8 hold. Then forε small enough we have

kcε,f−c0,fkf ≲ε¹², (6.8a) k∇(c_ε,f−c_0,f)kf ≲1, (6.8b) kcε,s−c0,skH¹(Ω_s)≲ε¹⁴. (6.8c) If additionally c0,f ∈H²(Ωf)then

kc_ε,f−c_0,fkf ≲ε³⁴, (6.9a) k∇(cε,f−c0,f)kf ≲ε¹⁴, (6.9b) kcε,s−c0,skH¹(Ωs)≲ε¹², (6.9c) i.e. the order is increased by ¹₄ compared toineqs.(6.8).

Proof. Set e_ε,f := c_ε,f −c_0,f and e_ε,s :=c_ε,s−c_0,s. By stability of c_ε,f and c_0,f from ineq. (6.4), respectivelyAssumption 6.2.2, ineq. (6.8b)is trivial. Let us outline the derivation of the other estimates in ineqs. (6.8): We use As-sumption 6.2.8 to establish the estimate for keε,fkf and with Lemma 6.2.10 we estimatekeε,fkξ inineq. (6.6)usingineq. (6.8a)andineq. (6.8b)to establish the bound foreε,s.

We modify a techniques from [Eck79, Chapter 6] to establish the estimate foreε,f. We have

a_ε,f(e_ε,f, ϕ_f) =−(∇c_0,f,∇ϕ_f)_f−(c_ε,f −c_ε,s, ϕ_f)_ξ (6.10) forϕf ∈Cf. By Assumption 6.2.2onc0,f and ineq. (6.4), the right-hand side is bounded:

−(∇c0,f,∇ϕf)f−(cε,f−cε,s, ϕf)ξ ≲k∇ϕfkf (6.11) with constant depending on the data. We choose as test functionϕ_f :=e^ψe_ε,f. For the left-hand side ofeq. (6.10)we must estimatea_ε,f(e_ε,f, ϕ_f)from below.

For the Laplace term there holds

(∇eε,f,∇(e^ψeε,f))f = (∇eε,f,e^ψ∇eε,f)f+ (∇eε,f,(∇e^ψ)eε,f)f

and for the second term, partial integration yields 2(∇e_ε,f,(∇e^ψ)e_ε,f)_f=−(e_ε,f,(∆e^ψ)e_ε,f)_f+ Z

∂Ω_f

∇e^ψ·n_f|e_ε,f|²do.

Writing min for min_x∈Ω

f and max for max_x∈Ω

f, this implies (∇e_ε,f,∇(e^ψe_ε,f))_f ≥min e^ψk∇e_ε,fk²f−¹₂max|∆e^ψ|ke_ε,fk²f

+¹₂ Z

∂Ω_f

∇e^ψ·nf|e_ε,f|²do. (6.12) The boundary term needs special care, since the sign of∇e^ψ·nf can be negative on∂Ωf\∂iΩf, in contrast to the similar integral withvf ·nf which occurred in the proof ofLemma 6.2.3. Let us assume that this situation occurs, i.e.

ess inf

∂Ωf\∂iΩf

∇e^ψ·nf <0,

otherwise we directly estimate this term inineq. (6.12). Using|nf|= 1we get 1

2 Z

∂Ω_f

∇e^ψ·nf|eε,f|²do≥ −¹₂max|∇e^ψ| Z

∂Ω_f

|eε,f|²do

≥ −^C₂ max|∇e^ψ|k∇eε,fk²f

whereC >0 is independent ofψ. Employing this inineq. (6.12), we conclude (∇eε,f,∇(e^ψeε,f))f ≥ min e^ψ−^C₂ max|∇e^ψ|

k∇eε,fk²f

−¹₂max|∆e^ψ|keε,fk²f.

If we rescaleψ7→ψ₀⁻¹ψ with some scalarψ0 >0, we see that the validity ofAssumption 6.2.8 is unaffected apart from the value ofν. Since

min e^ψ0ψ−^C₂ψ0max|∇ψe^ψ0ψ| →1 forψ0→0 we may assume that, after rescaling,ψis such that there holds

(∇eε,f,∇(e^ψeε,f))f ≥¹₂k∇eε,fk²f−¹₂max|∆e^ψ|keε,fk²f. (6.13) For the transport term ina_ε,f we have by a similar partial integration argument that the boundary integral vanishes by our assumptions onv_f

2(vf· ∇eε,f,e^ψeε,f)f=−(eε,f, vf· ∇ψe^ψeε,f)f+ Z

∂Ωf

vf·nfe^ψ|eε,f|²do

≥ −(eε,f, vf· ∇ψe^ψeε,f)f≥νmin e^ψkeε,fk²f (6.14) usingAssumption 6.2.8. Combining this estimate withineq. (6.13), we get

aε,f(eε,f,e^ψeε,f)≥¹₂k∇eε,fk²f + (_2ε^ν min e^ψ−¹₂max|∆e^ψ|)keε,fk²f. (6.15) Applyingineqs. (6.11)and(6.15)toeq. (6.10) we conclude

k∇eε,fk²f+ (¹_ε−C0)keε,fk²f ≲k∇(e^ψeε,f)k²f (6.16) with C₀ >0 independent of ε. For εsmall enough, ¹_ε−C₀ ≥ _2ε¹. Using the stability of e_ε,f and similar estimates as before imply k∇(e^ψe_ε,f)k²f ≲1 and henceineq. (6.8a)follows fromineq. (6.16). For the structure error,ineq. (6.8c), the arguments leading toineq. (6.6) in the proof ofLemma 6.2.7imply

keε,skH¹(Ωs)≲keε,fkL²(Γ).

Using the interpolation-type traceineq. (6.7)from Lemma 6.2.10we can con-cludeineq. (6.8c), since

ke_ε,skH¹(Ω_s)≲ke_ε,fkf¹²k∇e_ε,fkf¹² ≲ε¹⁴ (6.17) as ke_ε,fkf ≲ε¹² andk∇e_ε,fkf ≲1. This finishes the proof ifc_0,f ∈H¹(Ω_f).

If additionallyc_0,f ∈H²(Ω_f)we proceed as above and mainly improve the estimate of the right-hand side ofeq. (6.10). For the first term on the right of eq. (6.10) we have

−(∇c0,f,∇ϕf)f = (∆c0,f, ϕf)f−(∇c0,f ·n, ϕf)∂Ω_f\∂iΩ_f

≲kc0,fkH²(Ωf)(kϕfkf+kϕfkL²(∂Ωf\∂iΩf)).

Estimating the second term on the right ofeq. (6.10) as

−(c_ε,f−c_ε,s, ϕ_f)_ξ ≲kξkL^∞(Γ)(kc_ε,fkH¹(Ω_f)+kc_ε,skH¹(Ω_s))kϕ_fkL²(Γ)

we can hence conclude, using stability of cε,f andcε,s, and moving estimates for the data into the constant, that

−(∇c0,f,∇ϕf)f −(cε,f−cε,s, ϕf)ξ ≲kϕfkf+kϕfkL²(∂Ωf\∂iΩf). Withϕf=e^ψeε,f, as above, this leads to

−(∇c_0,f,∇ϕ_f)_f−(c_ε,f −c_ε,s, ϕ_f)_ξ≲ke_ε,fkf+ke_ε,fkL²(∂Ω_f\∂iΩ_f). (6.18) Using the coercivity bound for aε,f from ineq. (6.15) we can hence conclude from eq. (6.10), using arguments as in the derivation ofineq. (6.16), that

k∇eε,fk²f+¹_εkeε,fk²f ≲keε,fkf+keε,fkL²(∂Ω_f\∂iΩ_f). (6.19) For the second term on the right ofineq. (6.19)we use the trace ineq. (6.7)as in thee_ε,s-estimate so that there holds

keε,fkL²(∂Ω_f\∂iΩ_f)≲keε,fk_f¹²k∇eε,fk_f¹².

Inserting this estimate intoineq. (6.19)and rewriting the terms in preparation for the next step, we conclude

k∇eε,fk²f+¹_εkeε,fk²f ≤(Cε¹²)(¹_εkeε,fk²f)¹² + (Cε¹⁴)(¹_εkeε,fk²f)¹⁴(k∇eε,fk²f)¹⁴. Using the suggestive grouping of the terms on the right, we apply Young’s inequality with ¹₂ +¹₂ = 1for the first summand and with ¹₂+¹₄ +¹₄ = 1 for the second summand, rescaling such that

k∇eε,fk²f+¹_εkeε,fk²f≤Cε+_4ε¹keε,fk²f+Cε¹²+_4ε¹keε,fk²f+¹₂k∇eε,fk²f (6.20) which implies ineq. (6.9a)and ineq. (6.9b)since

k∇eε,fk²f+¹_εkeε,fk²f ≲ε+ε¹² ≲ε¹².

Forineq. (6.9c)we proceed as in the derivation ofineq. (6.17)in the low regu-larity case. With our improved estimates for e_ε,f this yields ineq. (6.9c)since

keε,skH¹(Ωs)≲keε,fk_f¹²k∇eε,fk_f¹² ≲ε^12(34+14)=ε¹². □

We now investigate the case−v_f· ∇ψ≥ν≥0with some functionν being zero in a small set. To outline the arguments: Proceeding as inTheorem 6.2.11 one getske_ε,fk²f,ν ≲εwherek · kf,ν is theν-weightedL²(Ω_f)-norm. Assuming that {ν = 0} is small andν well-behaved in a neighborhood, we can recover estimates without weight but with worse rate of convergence.

Assumption 6.2.12. Letψ be of the regularity as inAssumption 6.2.8with

−vf· ∇ψ≥ν ≥0

but now ν ∈C(Ωf)is allowed to be zero at some points in Ωf. Assume that for any0< δ < δ0there are subdomainsΩ^δ_f ⊂Ωf such that for any0< µ <1 there holds for allf ∈H¹(Ωf)

kfk²_L2(Ω_f\Ω^δ_f)≤µkfk²L²(Ω_f)+C(µ)δ²kfk²H¹(Ω_f) (6.21) with constant C(µ)>0depending onµbut not onδ. Furthermore,

ν₀(δ) :=ess inf

x∈Ω^δ_f

ν(x) is assumed to satisfy for someκ∈Nthe estimate

δ^κ

ν0(δ) ≲1 (6.22)

with constant independent ofδ.

One can think ofΩ^δ_f as a “nice” subset ofΩf with distanceδfrom{ν = 0}. The constantκmeasures the order at whichν vanishes asδ→0. To construct Ω^δ_f with the property fromineq. (6.21)the following lemma can be used.

Lemma 6.2.13. Letψandνbe as inAssumption 6.2.12. Furthermore, assume N :={x∈Ωf |ν(x) = 0} ⊂∂Ωf

andN 6=∅. If there is δ₀>0 such that

Λ :N×[0, δ0)→Ωf, (xN, d)7→xN−dnf(xN) is aC¹-diffeomorphism onto its image, we can choose

Ω^δ_f:= Ωf\Λ(N×[0, δ))

with 0 < δ < δ0 for the sets in Assumption 6.2.12 such that ineq.(6.21) is satisfied.

Proof. Letf ∈H¹(Ωf). Transformation into normal coordinates byΛyields Z

with constants independent ofδ. Proceeding as in the proof ofineq. (6.7)from Lemma 6.2.10we get

kfk²L²(N)≤ kfkL²(∂Ωf)≲kfk_f¹²kfk_H¹²1(Ω_f).

We can hence conclude that ineq. (6.21)holds since

kfk²_L2(Ω_f\Ω^δ_f)≲kfkf(δ²kfk²H¹(Ω_f))¹² +δ²kfk²H¹(Ω_f)

≤µkfk²f+C(µ)δ²kfk²H¹(Ω_f)

withC(µ)>0independent ofδ. □

Example 6.2.14. InΩf := (−^L₂,^L₂)×(0,1)withL >0consider the Poiseuille-type flow

v(x, y) =v0(y^µ(1−y)^µ,0)^T

forv0>0andµ∈N. ThenAssumption 6.2.12is satisfied with ψ(x) = (−1,0)^T·x

and κ =µ, supporting the interpretation that κ measures the rate at which

the flow vanishes.

Proof. The conditions ofLemma 6.2.13are satisfied with ν(x, y) =v₀y^µ(1−y)^µ≥0.

There holdsN = (0, L)×{0,1}andΛis a diffeomorphism for0< δ₀≤¹₂, with Ω^δ_f= (0, L)×(δ,1−δ).

Finally

ν₀(δ) = ess inf

(x,y)∈Ω^δ_f

ν(x) =v₀ ess inf

y∈(δ,1−δ)

y^µ(1−y)^µ=v₀δ^µ(1−δ)^µ which implies that the optimal choice forκisκ=µsince

δ^κ

ν0(δ) =δ^κ−µ 1

v0(1−δ)≲1

with constant independent ofδif and only if κ≥µ. □

Definition 6.2.15. For ν from Assumption 6.2.12, (·,·)f,ν := (ν¹²·, ν¹² ·)f

defines the weightedL²_ν(Ωf)-space with normk · kL²_ν(Ω_f)=k · kf,ν. Before proceeding to the convergence proof underAssumption 6.2.12, we investigate how a ν-weighted L²(Ωf) estimate and stability in H¹(Ωf) with respect to εyields non-weighted error estimates inL²(Ωf).

Lemma 6.2.16. LetAssumption 6.2.12be satisfied for someκ∈N. Then for any f ∈H¹(Ωf)there holds

kfk²f ≲δ^−κkfk²f,ν+δ²kfk²H¹(Ωf)

for any 0< δ < δ0 with constant independent ofδ.

Proof. For any0< δ < δ0 split kfk²f =

Z

Ω^δ_f

|f|²dx+ Z

Ω_f\Ω^δ_f

|f|²dx. (6.23) For the first term we useineq. (6.22)such that

Z

Ω^δ_f

|f|²dx≤ 1 ν₀(δ)

Z

Ω^δ_f

ν|f|²dx≲δ^−κ Z

Ω^δ_f

ν|f|²dx.

For the second term on the right ofeq. (6.23) we useineq. (6.21)withµ=¹₂: kfk²_L2(Ω_f\Ω^δ_f)≤ ¹₂kfk²f+Cδ²kfk²H¹(Ω_f).

Moving the first term on the right to the left-hand side ineq.(6.23)we conclude kfk²f ≲δ^−κkfk²f,ν+δ²kfk²H¹(Ωf). □

Theorem 6.2.17. Let c_ε,f and c_ε,s solve Problem 6.2.1, c_0,f and c_0,s solve Problem 6.2.5. Let assumption Assumption 6.2.12 hold. Then for ε small enough we have

kcε,f −c0,fkf ≲ε^2+κ1 , (6.24a) k∇(c_ε,f−c_0,f)kf ≲1, (6.24b) kcε,s−c0,skH¹(Ωs)≲ε^4+2κ1 (6.24c) withκ∈Nas given byAssumption 6.2.12. If additionallyc0,f ∈H²(Ωf)then

kcε,f −c0,fkf ≲ε^4+2κ3 , (6.25a) k∇(c_ε,f−c_0,f)kf ≲ε^4+2κ1 , (6.25b) kcε,s−c0,skH¹(Ωs)≲ε^2+κ1 . (6.25c) Proof. Inequality (6.24b) follows from stability of cε,f and c0,f in H¹(Ωf).

For ineq. (6.24a) we repeat the main steps of the proof of Theorem 6.2.11.

Inequality (6.11)for the right-hand side ofeq. (6.10)implies

a_ε,f(e_ε,f, ϕ_f) =−(∇c_0,f,∇ϕ_f)_f−(c_ε,f−c_ε,s, ϕ_f)_ξ ≲k∇ϕ_fkf (6.26) and for ϕf := e^ψeε,f there holds k∇ϕfkf ≲ 1. For the Laplace term in aε,f

the arguments leading toineq. (6.13)can be repeated verbatim, i.e.

(∇eε,f,∇(e^ψeε,f))f ≥¹₂k∇eε,fk²f−¹₂max|∆e^ψ|keε,fk²f. (6.27) Using the new assumptions, we can modify the arguments for the advection term fromineq. (6.14)to arrive at

2(vf· ∇eε,f,e^ψeε,f)f ≥min e^ψkeε,fk²f,ν. (6.28) Combiningineq. (6.27)andineq. (6.28)we have

aε,f(eε,f,e^ψeε,f)≥¹₂k∇eε,fk²f−¹₂max|∆e^ψ|keε,fk²f+_2ε¹ min e^ψkeε,fk²f,ν. (6.29)

In the current scenario we could move the |∆e^ψ|-term to the right-hand side of eq. (6.26). In preparation for the case c0,f ∈ H²(Ωf) we instead use Lemma 6.2.16 to absorb this term into the other two terms of ineq. (6.29).

By Lemma 6.2.16there exists C0>0such that

max|∆e^ψ|keε,fk²f ≤C0δ^−κkeε,fk²f,ν+C0δ²k∇eε,fk²f

holds for any0< δ < δ₀. This implies

aε,f(eε,f,e^ψeε,f)≥ ¹₂(1−C0δ²)k∇eε,fk²f+¹₂(¹_εmin e^ψ−C0δ^−κ)keε,fk²f,ν. and for δ := (2C0ε)^1κ(min e^ψ)^−κ1 and ε small enough such that δ < δ0 and C0δ²<¹₂ we conclude

a_ε,f(e_ε,f,e^ψe_ε,f)≥¹₄(k∇e_ε,fk²f+¹_εmin e^ψke_ε,fk²f,ν). (6.30) Using this coercivity-type estimate ineq. (6.26) yields byk∇ϕfkf ≲1that

k∇eε,fk²f+¹_εkeε,fk²f,ν ≲k∇ϕfkf ≲1 (6.31) which implies in particular

ke_ε,fk²f,ν ≲ε.

We useLemma 6.2.16to pass to the non-weightedL²-norm ofe_ε,f. Combining the previous estimate with the stability ofe_ε,f in H¹(Ω_f), we get

keε,fk²f ≲δ^−κkeε,fk²f,ν+δ²keε,fk²H¹(Ω_f)≲δ^−κε+δ²

for arbitrary0< δ < δ₀. With the Ansatzδ=ε^αforα >0the right-hand side is balanced ifα=_2+κ¹ , which yieldsineq. (6.24a). Inequality (6.24c)follows as in the proof ofTheorem 6.2.11.

In case of enhanced regularity,c0,f ∈H²(Ωf), we use the improved estimate from ineq. (6.18) from the proof of Theorem 6.2.11 for the right-hand side of eq. (6.26). Combined with the coercivity-type estimate from ineq. (6.30) for aε,f this implies

k∇e_ε,fk²f+¹_εke_ε,fk²f,ν ≲ke_ε,fkf+ke_ε,fkL²(∂Ω_f\∂iΩ_f). (6.32) For the first term on the right ofineq. (6.32)we have

keε,fkf ≲ε^−κα2 keε,fkf,ν+ε^αk∇eε,fkf

≲ε^1−κα2 (¹_εkeε,fk²f,ν)¹² +ε^αk∇eε,fkf

≤Cε^1−κα+_4ε¹keε,fk²f,ν+Cε^2α+¹₆k∇eε,fk²f

for anyα >0byLemma 6.2.16withδ=ε^αand Young’s inequality. With the choice α= _2+κ¹ theε-powers are balanced, implying

keε,fkf ≤Cε^2+κ2 +_4ε¹keε,fk²f,ν+¹₆k∇eε,fk²f. (6.33) For the second term on the right ofineq. (6.32)we again use the interpolation-type trace estimate fromineq. (6.7). Withαas above then

keε,fkL²(∂Ω_f\∂iΩ_f)≲keε,fkf¹²k∇eε,fkf¹² ≲ε^α2 ε^−αkeε,fkf

¹₂

k∇eε,fkf¹².

Using Young’s inequality with ¹₂+¹₄+¹₄= 1 we arrive at

ke_ε,fkL²(∂Ω_f\∂iΩ_f)≤Cε^α+µε^−2αke_ε,fk²f+¹₆k∇e_ε,fk²f (6.34) with free parameterµ >0. This parameter is chosen such that application of Lemma 6.2.16withδ=ε^αyields

µke_ε,fk²f ≤ε^1−κα_4ε¹ke_ε,fk²f,ν+ε^2α1₆k∇e_ε,fk²f. Sinceα= _2+κ¹ the exponents forεon the right are equal and

µε^−2αkeε,fk²f ≤ _4ε¹keε,fk²f,ν+¹₆k∇eε,fk²f. Inserting this estimate inineq. (6.34)we arrive at

ke_ε,fkL²(∂Ω_f\∂iΩ_f)≤Cε^2+κ1 +_4ε¹ke_ε,fk²f,ν+¹₃k∇e_ε,fk²f. (6.35) Combiningineqs. (6.33)and(6.35)in the right-hand side ofineq. (6.32)we get

k∇eε,fk²f+¹_εkeε,fk²f,ν≤Cε^2+κ2 +Cε^2+κ1 +_2ε¹keε,fk²f,ν+¹₂k∇eε,fk²f

and hence, estimating the higher orderεterm, we arrive atineq. (6.25b), since k∇e_ε,fk²f+¹_εke_ε,fk²f,ν≲ε^2+κ1 . (6.36) The unweighted L²(Ω_f) estimate from ineq. (6.25a) follows as above using Lemma 6.2.16 withδ=ε^α forα= _2+κ¹ . The finalineq. (6.25c) fore_ε,s again

follows as in the proof ofTheorem 6.2.11. □

Remark 6.2.18. Setting formally κ = 0 in Theorem 6.2.17 we recover the

estimates fromTheorem 6.2.11.

Im Dokument Temporal Multiscale Methods for a Model of Atherosclerosis (Seite 150-161)