HEAT FLOW PROBLEMS ACROSS FRACTAL LAYERSmaria.lancia/seminarionapoli2011.pdf · HEAT FLOW PROBLEMS ACROSS FRACTAL LAYERS M.Rosaria Lancia Dipartimento di Scienze di Base per le Applicazioni

HEAT FLOW PROBLEMS ACROSSFRACTAL LAYERS

M.Rosaria Lancia

Dipartimento di Scienze di Base per le Applicazioni e l’IngegneriaSezione di Matematica

Università degli Studi di Roma “La Sapienza"

New Function Spaces in PDESand

Harmonic AnalysisNaples, MAY 31-JUNE 4, 2011

M.Rosaria Lancia HEAT FLOW PROBLEMS ACROSS FRACTAL LAYERS

HEAT FLOW PROBLEMS ACROSSFRACTAL LAYERS

M.Rosaria Lancia

Dipartimento di Scienze di Base per le Applicazioni e l’IngegneriaSezione di Matematica

Università degli Studi di Roma “La Sapienza"

New Function Spaces in PDESand

Harmonic AnalysisNaples, MAY 31-JUNE 4, 2011


THE MODEL: SMOOTH LAYER ( P.Huy, E. Sanchez Palencia 1974)

Q = (−1, 1)2 × (0, 1)S = (−1, 1)2 × 1

2 ,Ωi = S × ei ;

ei = ( 12 −

1i ,

12 ).

0

0.5

1

Q=[−1;1]2x[0,1]

ei

Ωi

(Li)

cQut − ηQ∆u(P) = f (P) in (Q \ Ωi),ciut − ηi∆u(P) = 0 in Ωi ,u(P) = 0 on ∂Q,u(P)|Q\Ωi = u(P)|Ωi on ∂Ωi ,

ηQ∂u(P)

∂n |Q\Ωi = ηi∂u(P)

∂n |Ωi on ∂Ωiu(P)|t=0 = 0

f given function, ηi , ηQ thermal conductivity, ci , cQ specific heat.HP: limi→∞ ηi

1i = ηS whereηS = 0, eitherηS = ∞orηS > 0,

limi→∞

ci1i

= 0 or 6= 0


THE MODEL: SMOOTH LAYER ( P.Huy, E. Sanchez Palencia 1974)

Q = (−1, 1)2 × (0, 1)S = (−1, 1)2 × 1

2 ,Ωi = S × ei ;

ei = ( 12 −

1i ,

12 ).

0

0.5

1

Q=[−1;1]2x[0,1]

ei

Ωi

(Li)

cQut − ηQ∆u(P) = f (P) in (Q \ Ωi),ciut − ηi∆u(P) = 0 in Ωi ,u(P) = 0 on ∂Q,u(P)|Q\Ωi = u(P)|Ωi on ∂Ωi ,

ηQ∂u(P)

∂n |Q\Ωi = ηi∂u(P)

∂n |Ωi on ∂Ωiu(P)|t=0 = 0

f given function, ηi , ηQ thermal conductivity, ci , cQ specific heat.HP: limi→∞ ηi

1i = ηS whereηS = 0, eitherηS = ∞orηS > 0,

limi→∞

ci1i

= 0 or 6= 0


LIMIT PROBLEMS 3D

L(ηS = 0)

−cQut − ηQ∆u(P) = f (P) in Q,u(P) = 0 on ∂Q,

L(ηS = ∞)

−cQut − ηQ∆u(P) = f (P) in Q \ S,

u1(P) = u2(P) = 0 on S,u = 0 on∂Q

ui = u|Qi , i = 1, 2.


LIMIT PROBLEMS 3D ηS < ∞limi→∞ ci

1i = 0

L

cQut(t , P)− ηQ∆u(t , P) = f (t , P) on [0, T ]×Qi ,

−ηS∆Su(t , P) =[

∂u(t ,P)∂n

]on [0, T ]× S,

u(t , P) = 0 on [0, T ]× ∂Q,

u1(t , P) = u2(t , P) on [0, T ]× S,u(0, P) = 0 on Q

limi→∞ ci1i = cS 6= 0

L


cSut(t , P)− ηS∆Su(t , P) =[

∂u(t ,P)∂n

]on [0, T ]× S,

u(t , P) = 0 on [0, T ]× ∂Q,



LIMIT PROBLEMS 3D ηS < ∞limi→∞ ci

1i = 0

L


−ηS∆Su(t , P) =[

∂u(t ,P)∂n

]on [0, T ]× S,

u(t , P) = 0 on [0, T ]× ∂Q,


limi→∞ ci1i = cS 6= 0

L


cSut(t , P)− ηS∆Su(t , P) =[

∂u(t ,P)∂n

]on [0, T ]× S,

u(t , P) = 0 on [0, T ]× ∂Q,



MOTIVATIONS

2ND ORDER TRANSMISSION PROBLEMS occur in manyfields:

Electrostatics, Magnetostatics: P.Huy, E.Sanchez Palencia

Hydraulic facturing: Cannon Meyer

Irrigation problems: Morel

INTERESTING CASE : Fractal interfaces ==> Largesurfaces vs small volumes


TRANSMISSION PROBLEMS FRACTAL LAYERLayer condition of order 2prefractal fractal

Q1

Q2

S

Sh = Fh×I, Df = 2; S = F×I 2 < Df < 3

F : snowflake, I = [0, 1]

ut(t , P)−∆u(t , P) = f (t , P) in [0, T ]×Q i ,

ut(t , P)− ηS∆Su(t , P) =[

∂u(t,P)∂n

]on [0, T ]× S,

u(t , P) = 0 on [0, T ]× ∂Q,u1(t , P) = u2(t , P) on [0, T ]× S,u(t , P) = 0 on [0, T ]× ∂Su(0, P) = 0 on Q,


THE EXTENSION TO THE FRACTAL CASERequired KEY TOOLS:

Sobolev-type functions spaces on fractal setsTrace theorems for domains with fractal boundariesGreen formulas for domains with fractal boundariesDefinition of normal derivative on fractal sets

THE PRESENCE OF THE TIME DERIVATIVE ON S AND Sh

Suitable Functional settingVarying Hilbert spaces


RESULTS (2002-2011)Elliptic Case 2D and 3D: Variational Methods (L., L.-Vivaldi)Homogeneization Approach for "THICK" prefractal or fractallayers (Mosco-Vivaldi 2007-2010, L.-Mosco-Vivaldi 2008)Numerical approximation by FEM (Vacca 2005)Parabolic case ci

i → 0Semigroup approach (L.-Vernole 2006)Numerical approximation by FEM + FINITE DIFFERENCESCHEMES , error estimates (L.-Vacca 2008)Parabolic case ci

i → cS 6= 0 Koch-pipe (L.-Vernole 2010);Error estimates and numerical simulations by FEM + FINITEDIFFERENCE SCHEMES,(Cefalo-L.-Dell’Acqua, 2011);Semi-linear case evolution problems across fractal layers(L.-Vernole 2011), Insulating thin layers (L.-Vernole, 2011)Numerical approximation by FEM for Koch-type curves:(Wasyk,2007)(Evans,2008-2011)


RESULTS (2002-2011)Elliptic Case 2D and 3D: Variational Methods (L., L.-Vivaldi)Homogeneization Approach for "THICK" prefractal or fractallayers (Mosco-Vivaldi 2007-2010, L.-Mosco-Vivaldi 2008)Numerical approximation by FEM (Vacca 2005)Parabolic case ci

i → 0Semigroup approach (L.-Vernole 2006)Numerical approximation by FEM + FINITE DIFFERENCESCHEMES , error estimates (L.-Vacca 2008)Parabolic case ci

i → cS 6= 0 Koch-pipe (L.-Vernole 2010);Error estimates and numerical simulations by FEM + FINITEDIFFERENCE SCHEMES,(Cefalo-L.-Dell’Acqua, 2011);Semi-linear case evolution problems across fractal layers(L.-Vernole 2011), Insulating thin layers (L.-Vernole, 2011)Numerical approximation by FEM for Koch-type curves:(Wasyk,2007)(Evans,2008-2011)


The SNOWFLAKE-type F


THE Koch-type pipe S = F × I, I = [0, 1]


The SNOWFLAKE F AND THE FRACTAL LAYER S = F × I, I = [0, 1]The snowflake

x1

x2

x3

x4

x5

x6

K1

K2

K3 F =⋃

i=1,2,3 Ki =⋃

i=4,5,6 KisptµF = F ,µF = (Hdf (F ))−1Hdf |F ,where df = ln 4

ln 3

The Fractal Layer

Q1

Q2

S

P ∈ S , P = (x , y);x = (x1, x2), x = (x1, x2) ∈ F ,y ∈ I.

spt(g) = Sdg = dµF × dy ; dydy Lebesgue meas. on Id = df + 1 = log 12

log 3 S is ad–set with d = df + 1.


The SNOWFLAKE F AND THE FRACTAL LAYER S = F × I, I = [0, 1]The snowflake

x1

x2

x3

x4

x5

x6

K1

K2

K3 F =⋃

i=1,2,3 Ki =⋃

i=4,5,6 KisptµF = F ,µF = (Hdf (F ))−1Hdf |F ,where df = ln 4

ln 3

The Fractal Layer

Q1

Q2

S

P ∈ S , P = (x , y);x = (x1, x2), x = (x1, x2) ∈ F ,y ∈ I.

spt(g) = Sdg = dµF × dy ; dydy Lebesgue meas. on Id = df + 1 = log 12

log 3 S is ad–set with d = df + 1.


The PREFRACTAL LAYER

K

P

Sj

xy

I

S

Sh polyhedral-type surface.Sh = Fh × I,h = 1, 2, . . .,` : arc–length on each edgeof Fh.dσ surface measure on eachface S(j)

h of Sh,

dσ = d`dy .

L2(Q, mh), the Lesb. space w.r.t. the measure mh = dQ + δhdσ

where δh is a positive factor.

L2(Q, m) the Lesb. space w.r.t. the measure m = dQ + dg


The PREFRACTAL LAYER

K

P

Sj

xy

I

S

Sh polyhedral-type surface.Sh = Fh × I,h = 1, 2, . . .,` : arc–length on each edgeof Fh.dσ surface measure on eachface S(j)

h of Sh,

dσ = d`dy .

L2(Q, mh), the Lesb. space w.r.t. the measure mh = dQ + δhdσ

where δh is a positive factor.

L2(Q, m) the Lesb. space w.r.t. the measure m = dQ + dg


THE FRACTAL LAPLACIAN ∆S

The energy form on S:ES[u] =

∫I EF [u]dy +

∫F

∫I |Dyu|2dyµF (dx).

EF energy form of F with domain D(F ) dense inL2(F ) = L2(F , µF (dx)).The form ES is defined for u ∈ D(S) where

D(S) = C0(S) ∩ L2(0, 1;D(F )) ∩ H10 (0, 1; L2(F ))

‖·‖

w.r.t. the intrinsic norm

‖u‖D(S) = (ES[u] + ‖u‖2L2(S,g))

12

The operator ∆S on L2(S, g)—with domain D(∆S) ⊆ D(S)dense in L2(S, g)— is defined as

ES(u, v) = −∫

S(∆Su)vdm, u ∈ D(∆S), v ∈ D(S).


THE FUNCTIONAL SETTING IN THE FRACTAL CASE

S : L2(S, g) The Lesb. space w.r.t. the measure dg

Q : L2(Q, m) The Lesb. space w.r.t. the measure m = dQ + dg

Proposition

ES with domain D(S) is a regular Dirichlet form in L2(S, g) andthe space D(S) is a Hilbert space under the intrinsic norm.


ES(u, v) = −∫

S(∆Su)vdg, u ∈ D(∆S), v ∈ D(S).

L2(Q, m) := u : Q → R : (∫

Q |u|2dQ +

∫S |u|

2dg)1/2 < ∞M.Rosaria Lancia HEAT FLOW PROBLEMS ACROSS FRACTAL LAYERS




Proposition



ES(u, v) = −∫

S(∆Su)vdg, u ∈ D(∆S), v ∈ D(S).

L2(Q, m) := u : Q → R : (∫

Q |u|2dQ +

∫S |u|





Proposition



ES(u, v) = −∫

S(∆Su)vdg, u ∈ D(∆S), v ∈ D(S).

L2(Q, m) := u : Q → R : (∫

Q |u|2dQ +

∫S |u|


THE FRACTAL ENERGY AND THE ASSOCIATED SEMIGROUP

E [u] =

∫Q|Du|2dQ + ηSES[u|S], u ∈ V (Q, S)

V (Q, S) =

u ∈ H10 (Q) : u|S ∈ D(S)

; ηS > 0.

Prop. 2 (E , V (Q, S)) is a regular Dirichlet form in L2(Q, m), V (Q, S)

is a Hilbert space, ‖u‖V (Q,S) = (E [u])12 .

∃ a unique self-adj. non pos. operator A on L2(Q, m) with domainD(A) ⊆ V (Q, S) dense in L2(Q, m) such that

E(u, v) = −∫

QAu v dQ, u ∈ D(A), v ∈ V (Q, S)

Prop. 3Let T (t)t≥0 be the semigroup generated by the operator Aassociated to the energy form in E . Then T (t)t≥0 is an analyticcontraction semigroup in L2(Q, m).



E [u] =


V (Q, S) =

u ∈ H10 (Q) : u|S ∈ D(S)

; ηS > 0.




E(u, v) = −∫





E [u] =


V (Q, S) =

u ∈ H10 (Q) : u|S ∈ D(S)

; ηS > 0.




E(u, v) = −∫




THE PREFRACTAL ENERGY AND THE ASSOCIATED SEMIGROUP

E (h)[u] =

∫Q|Du|2dQ + ESh [u|Sh ], u ∈ V (Q, Sh)

V (Q, Sh) =

u ∈ H10 (Q) : u|Sh ∈ H1

0 (Sh)

.where the layer energyESh [u] = σ1

h∫

I

(∫Fh|Dù|2d`

)dy + σ2

h∫

Fh

(∫I |Dyu|2dy

)d`.

Prop.4 (E (h), V (Q, Sh)) is a regular Dirichlet form in L2(Q, mh)

and V (Q, Sh) is a Hilbert space, ‖u‖V (Q,Sh) = (E (h)[u])12 .

......Let Ah with domain D(Ah) ⊆ V (Q, Sh) dense in L2(Q, mh)be the generator of E (h)(u, v)Prop. 5Let Th(t)t≥0 be the semigroup generated by theoperator Ah associated to the energy form E (h). Th(t)t≥0 isan analytic contraction semigroup in L2(Q, mh).



E (h)[u] =


V (Q, Sh) =

u ∈ H10 (Q) : u|Sh ∈ H1

0 (Sh)


h∫

I

(∫Fh|Dù|2d`

)dy + σ2

h∫

Fh

(∫I |Dyu|2dy

)d`.






E (h)[u] =


V (Q, Sh) =

u ∈ H10 (Q) : u|Sh ∈ H1

0 (Sh)


h∫

I

(∫Fh|Dù|2d`

)dy + σ2

h∫

Fh

(∫I |Dyu|2dy

)d`.





ABSTRACT CAUCHY PROBLEMS

(P)

du(t)dt = Au(t) + f (t), 0 ≤ t ≤ T

u(0) = 0

and for every h

(Ph)

duh(t)dt = Ah uh(t) + fh(t), 0 ≤ t ≤ T

uh(0) = 0

A : D(A) ⊂ L2(Q, m) → L2(Q, m) andAh : D(Ah) ⊂ L2(Q, mh) → L2(Q, mh) are the generatorsassociated respectively to the energy form E and the energyforms E (h), T is a fixed positive real number, andf ∈ Cθ([0, T ], L2(Q, m)) ,fh ∈ Cθ([0, T ], L2(Q, mh)).


ABSTRACT CAUCHY PROBLEMS

(P)

du(t)dt = Au(t) + f (t), 0 ≤ t ≤ T

u(0) = 0

and for every h

(Ph)

duh(t)dt = Ah uh(t) + fh(t), 0 ≤ t ≤ T

uh(0) = 0

A : D(A) ⊂ L2(Q, m) → L2(Q, m) andAh : D(Ah) ⊂ L2(Q, mh) → L2(Q, mh) are the generatorsassociated respectively to the energy form E and the energyforms E (h), T is a fixed positive real number, andf ∈ Cθ([0, T ], L2(Q, m)) ,fh ∈ Cθ([0, T ], L2(Q, mh)).


EXISTENCE AND UNIQUENESS RESULTS ( L.-Vernole 2010)

Theorem

Let 0 < θ < 1, f ∈ Cθ([0, T ], L2(Q, m)) and let

u(t) =

∫ t

0T (t − s) f (s) ds,

where T (t) is the analytic semigroup generated by A. Then u isthe unique “strict” solution of (P) i.e.

u ∈ C1([0, T ]; L2(Q, m)) ∩ C([0, T ]; D(A))

du(t)dt = Au(t) + f for every t ∈ [0, T ]; and u(0) = 0.

Furthermore, there exists c such that‖u‖C1([0,T ],L2(Q,m)) + ‖u‖C0([0,T ],D(A)) ≤ c‖f‖Cθ([0,T ],L2(Q,m))


EXISTENCE AND UNIQUENESS RESULTS ( L.-Vernole 2010)

Theorem

Let 0 < θ < 1, f ∈ Cθ([0, T ], L2(Q, m)) and let

u(t) =

∫ t

0T (t − s) f (s) ds,

where T (t) is the analytic semigroup generated by A. Then u isthe unique “strict” solution of (P) i.e.

u ∈ C1([0, T ]; L2(Q, m)) ∩ C([0, T ]; D(A))

du(t)dt = Au(t) + f for every t ∈ [0, T ]; and u(0) = 0.

Furthermore, there exists c such that‖u‖C1([0,T ],L2(Q,m)) + ‖u‖C0([0,T ],D(A)) ≤ c‖f‖Cθ([0,T ],L2(Q,m))


Theorem

Let 0 < θ < 1, fh ∈ Cθ([0, T ], L2(Q, mh)) and let

uh(t) =

∫ t

0Th(t − s) fh(s) ds for every h

where Th(t) is the analytic semigroup generated by Ah. Thenuh is the unique “strict” solution of (Ph) i.e.

uh ∈ C1([0, T ]; L2(Q, mh)) ∩ C([0, T ]; D(Ah))

duh(t)dt = Ah uh(t) + fh for every t ∈ [0, T ], and uh(0) = 0.

Furthermore there exists c, independent from h, such that‖uh‖C1([0,T ],L2(Q,mh))

+ ‖uh‖C0([0,T ],D(Ah))≤ c‖fh‖Cθ([0,T ],L2(Q,mh))


Theorem

Let 0 < θ < 1, fh ∈ Cθ([0, T ], L2(Q, mh)) and let

uh(t) =

∫ t

0Th(t − s) fh(s) ds for every h

where Th(t) is the analytic semigroup generated by Ah. Thenuh is the unique “strict” solution of (Ph) i.e.

uh ∈ C1([0, T ]; L2(Q, mh)) ∩ C([0, T ]; D(Ah))

duh(t)dt = Ah uh(t) + fh for every t ∈ [0, T ], and uh(0) = 0.

Furthermore there exists c, independent from h, such that‖uh‖C1([0,T ],L2(Q,mh))

+ ‖uh‖C0([0,T ],D(Ah))≤ c‖fh‖Cθ([0,T ],L2(Q,mh))


THE CONVERGENCE OF HILBERT SPACES, ( Kuwae-Shyoia (2003))

SetHh =

L2(Q, mh)

h∈N

, H = L2(Q, m)

dmh = dQ + δhdσ dm = dQ + dg,

Theorem

Let δh = (31−df )h. The sequence of Hilbert spaces Hhh∈Nconverges to the Hilbert space H in the sense of K-S.

Def.The sequence of Hilbert spaces Hhh∈N converges to theHilbert space H if ‖u‖Hh → ‖u‖H , h →∞ forany u ∈ C(Q)

Prop. uh ∈ Hh converges to u ∈ H iffi) ‖uh‖Hh → ‖u‖H , h →∞ii) (uh, φ)Hh → (u, φ)H ,∀φ ∈ C(Q)



SetHh =

L2(Q, mh)

h∈N

, H = L2(Q, m)


Theorem






SetHh =

L2(Q, mh)

h∈N

, H = L2(Q, m)


Theorem





The M-CONVERGENCE (U.Mosco 1969), Kuwae-Shyoia (2003))We extend the forms E(·, ·) and E (h)(·, ·) on the spacesL2(Q, m) and L2(Q, mh) respectively by defining

E(u, u) = +∞ for every u ∈ L2(Q, m) \ V (Q, S)

E (h)(u, u) = +∞ for every u ∈ L2(Q, mh) \ V (Q, Sh) .

Definition 1 A sequence of forms E (h)(·, ·) M-converges to aform E(·, ·) if(a) For every vh ∈ Hh converging weakly to u ∈ H

lim E (h)(vh, vh) ≥ E(u, u) , as h →∞.

(b) For every u ∈ H there exists wh with wh ∈ Hh convergingstrongly to u ∈ H such thatlimE (h)(wh, wh) ≤ E(u, u) , as h →∞.


Definition 2 The sequence of forms E (h)(·, ·) is asymptoticallycompact if every sequence uh s.t. uh ∈ Hh with

lim(E (h)(uh, uh) + ‖uh‖2Hh

) < ∞

has a subsequence strongly convergent.Prop.6 The sequence of forms E (h)(uh, uh) is asymptoticallycompact in L2(Q).Remark 1 As the sequence of forms E (h)(uh, uh) is asymptoticallycompact in L2(Q), M–convergence is equivalent to theΓ–convergence , thus we can take in (a) vh strongly converging to uin L2(Q).


M-K-S CONVERGENCE OF ENERGIES

Theorem

(L.- Vernole 2010) Let δh = (31−df )h, σ1h = ηS(δh)

−1 andσ2

h = ηSδh, then the sequence of forms

E (h)

M-K-Sconverges to the form E.

Corollary

Let G(h)α and Gα the resolvents associated to E (h) and E

respectively, then for every α > 0 the sequence

Ghα

converges to the operator Gα in the sense of K-S.

TheoremThe sequence of semigroups Th(t) associated with the formE (h) converges to the semigroup T (t) associated with the formE in the sense of K-S for every t>0.



Theorem


−1 andσ2


E (h)


Corollary



Ghα





Theorem


−1 andσ2


E (h)


Corollary



Ghα




Convergence of solutions

TheoremLet u and uh be the solutions of problems (P) and (Ph). Let δhas before. If for every t ∈ [0, T ], fh(t) strongly converges tof (t) in H and there exists a constant c such that

‖fh‖Cϑ([0,T ];Hh)< c for every h ∈ N, (1)

We have:uh ∈ Hh converges to u(t) ∈ H for every fixed t ∈ [0, T ]

duh(t)dt weakly converges to du(t)

dt in L2([0, T ]×Q, dtxdQ)

duh(t)dt ∈ L2([0, T ]× Sh, dtxδhdσ) weakly converges to

du(t)dt ∈ L2([0, T ]× S, dtxdg)

uh converges to u in L2([0, T ]; H10 (Q))


STRONG FORMULATIONS

TheoremLet u be the solution of problem (P). Then we have for everyfixed t ∈ [0, T ]

ut(t , P)−4u(t , P) = f (t , P) for a.e. P ∈ Qi∂ui

∂ni∈

((B2,2

β,0)(S))′

, β = df2 ,

u(t , P) = 0 for P ∈ ∂Q

〈ut , z〉(D(S))′D(S) − ηS〈4Su|S, z〉(D(S))′D(S) =−⟨[

∂u∂n

], z

⟩(D(S))′D(S)

+ 〈f , z〉(D(S))′D(S)

where ui = u|Qi , ∂ui

∂ni, i = 1, 2 is the inward “normal derivative”,[

∂u∂n

]= ∂u1

∂n1+ ∂u2

∂n2and 4S is the fractal Laplacian.

tools :Besov spaces :Jonsson-Wallin 1984, 1995; Triebel 1997Green formulas for Fractal Domains:L. 2002-2003


STRONG FORMULATIONS

TheoremLet u be the solution of problem (P). Then we have for everyfixed t ∈ [0, T ]

ut(t , P)−4u(t , P) = f (t , P) for a.e. P ∈ Qi∂ui

∂ni∈

((B2,2

β,0)(S))′

, β = df2 ,

u(t , P) = 0 for P ∈ ∂Q

〈ut , z〉(D(S))′D(S) − ηS〈4Su|S, z〉(D(S))′D(S) =−⟨[

∂u∂n

], z

⟩(D(S))′D(S)

+ 〈f , z〉(D(S))′D(S)

where ui = u|Qi , ∂ui

∂ni, i = 1, 2 is the inward “normal derivative”,[

∂u∂n

]= ∂u1

∂n1+ ∂u2

∂n2and 4S is the fractal Laplacian.

tools :Besov spaces :Jonsson-Wallin 1984, 1995; Triebel 1997Green formulas for Fractal Domains:L. 2002-2003


Theorem

Let uh be the solution of problem (Ph). Then we have, for everyfixed t ∈ [0, T ],

(uih)t = ∆ui

h + f in L2(Qi), i = 1, 2 j)δhut −∆Shuh = −[∂uh

∂n ] + δhfh in L2(Sh) jj)u = 0 in H

12 (∂Q) jjj)

u1 = u2 in H1(Sh) jv)

u = 0 in H12 (∂Sh) v).

uih = uh|Qi

h,[

∂uh∂n

]=

∂u1h

∂n1+

∂u2h

∂n2, 4Sh piece-wise tangential

Laplacian associated to the Dirichlet form ESh .tools :

Kondrate’ev 1967, Nazarov, Plamenevsky 1974,JerisonKenig 1991,1995Green formulas for Lipschitz domains: Baiocchi, Capelo1984


Theorem

Let uh be the solution of problem (Ph). Then we have, for everyfixed t ∈ [0, T ],

(uih)t = ∆ui

h + f in L2(Qi), i = 1, 2 j)δhut −∆Shuh = −[∂uh

∂n ] + δhfh in L2(Sh) jj)u = 0 in H

12 (∂Q) jjj)

u1 = u2 in H1(Sh) jv)

u = 0 in H12 (∂Sh) v).

uih = uh|Qi

h,[

∂uh∂n

]=

∂u1h

∂n1+

∂u2h

∂n2, 4Sh piece-wise tangential

Laplacian associated to the Dirichlet form ESh .tools :

Kondrate’ev 1967, Nazarov, Plamenevsky 1974,JerisonKenig 1991,1995Green formulas for Lipschitz domains: Baiocchi, Capelo1984


REGULARITY RESULTS ( L.2003, L.-Vernole 2006)

TheoremFor every fixed t ∈ [0, T ],u1

h ∈ Hs1(Q1h), s1 < 8

5 , u2h ∈ Hs2(Q2

h), s2 < 74 and u ∈ C(Q).

tools:

r

P

r^1

Q1h =⇒ θ1 = 2

3π or 53π;

Q2h =⇒ θ2 = 1

3π or 43π

rα1Dβw1h ∈ L2(Q1

h), |β| = 2, α1 >25


h), |β| = 2, α2 >14

Interpolation Results =⇒ w ih ∈ H2−αi (Qi

h)




h ∈ Hs1(Q1h), s1 < 8

5 , u2h ∈ Hs2(Q2

h), s2 < 74 and u ∈ C(Q).

tools:

r

P

r^1

Q1h =⇒ θ1 = 2

3π or 53π;

Q2h =⇒ θ2 = 1

3π or 43π


h), |β| = 2, α1 >25


h), |β| = 2, α2 >14


h)




h ∈ Hs1(Q1h), s1 < 8

5 , u2h ∈ Hs2(Q2

h), s2 < 74 and u ∈ C(Q).

tools:

r

P

r^1

Q1h =⇒ θ1 = 2

3π or 53π;

Q2h =⇒ θ2 = 1

3π or 43π


h), |β| = 2, α1 >25


h), |β| = 2, α2 >14


h)


Weak Convergence of Normal derivatives

Theorem(L. Vernole 2010)For every fixed t ∈ [0, T ],both sides of the transmissioncondition converge to the corresponding terms in the dualspace of V (Q, S) :

(j)∫

Sh[∂uh

∂n ] v dσ →< [∂u∂n ], v >(D(S))′,D(S) ∀v ∈ V (Q, S).

(jj)∫

Sh∆Sh uhvdσ →< ∆Su, v >(D(S))′,D(S) ∀v ∈ V (Q, S).


NUMERICAL 2D APPROXIMATION (Cefalo,Dell’Acqua, L., 2011)The MODEL PROBLEM (Pn); n fixed

Kn = Pre−fractal curve

ρn = 1

f ∈ Cδ([0, T ], L2(Q, m))

dm = dxdy + ds

ut(t , P)−∆u(t , P) = f (t , P) in [0, T ]× Ωin,

ut(t , P)− ρn∆Knu(t , P) =[

∂u(t ,P)∂n

]+ f on [0, T ]× Kn,

u(t , P) = 0 on [0, T ]× ∂Ω,

u1(t , P) = u2(t , P) on [0, T ]× Kn,u(t , P) = 0 on [0, T ]× ∂Knu(0, P) = 0 on Ω,


STRONG INTERPRETATION

Theorem

Let un be the solution of problem (Pn). For every fixed t ∈ [0, T ],dun(t ,P)

dt −4un(t , P) = f (t , P), for P ∈ Ωin, a.e. i = 1, 2

∂uin

∂νi∈ L2(Kn), i = 1, 2

dundt −4Knun|Kn =

[∂un∂ν

]+ f , in L2(Kn) (∗)

un(t , P) = 0, for P ∈ ∂Ω

(2)

where uin = u|Ωi

n,[

∂un∂ν

]= ∂u1

n∂ν1

+ ∂u2n

∂ν2is the jump of the normal

derivatives across Kn, νi , i = 1, 2, is the inward normal vectorsand 4Kn is the piece-wise tangential Laplacian associated tothe Dirichlet form EKn . Moreover∂ui

n∂νi

∈ C([0, T ]; L2(Kn)), i = 1, 2. Condition (∗) =⇒ un ∈ H2(Kn)


For every t ∈ [0, T ],

duin(t ,P)dt = ∆ui

n + f , in L2(Ωin), i = 1, 2 j)

ρ−1n

dundt − ρn∆Knun = [∂un

∂ν ] + ρ−1n f , in L2(Kn) jj)

un = 0, in H12 (∂Ω) jjj)

u1n = u2

n , in H1(Kn) jv)

un = 0, in H12 (∂Kn) v).

In particular from condition jj) follows un ∈ H2(Kn).


REGULARITY RESULTS

Theorem

For every fixed t ∈ [0, T ], u1n ∈ H2,α1(Ω1

n), α1 > 25 ,

u2n ∈ H2,α2(Ω2

n), α2 > 14 , un ∈ C(Ω), un|Kn ∈ H2(Kn).

A B

Ω1n =⇒ θ1 = 2

3π or 53π;

Ω2n =⇒ θ2 = 1

3π or 43π

H2,αi (Ωin) = v ∈ H1(Ωi

n) : rαi · Dβv ∈ L2(Ωin), |β| = 2

r is the distance from the vertices of Kn whose angles are"reentrant".Equipped with the following norm

‖ v ‖H2,αi (Ωin)

:=‖ v ‖2

1,Ωin

+∑

|β|=2 ‖ rαi · Dβv ‖20,Ωi

n

1/2.

and H2(Kn) := v ∈ H1(Kn) : v |M ∈ H2(M),∀Msegment ofKn.


REGULARITY RESULTS

Theorem


n), α1 > 25 ,

u2n ∈ H2,α2(Ω2

n), α2 > 14 , un ∈ C(Ω), un|Kn ∈ H2(Kn).

A B

Ω1n =⇒ θ1 = 2

3π or 53π;

Ω2n =⇒ θ2 = 1

3π or 43π


n) : rαi · Dβv ∈ L2(Ωin), |β| = 2



:=‖ v ‖2

1,Ωin

+∑

|β|=2 ‖ rαi · Dβv ‖20,Ωi

n

1/2.



REGULARITY RESULTS

Theorem


n), α1 > 25 ,

u2n ∈ H2,α2(Ω2

n), α2 > 14 , un ∈ C(Ω), un|Kn ∈ H2(Kn).

A B

Ω1n =⇒ θ1 = 2

3π or 53π;

Ω2n =⇒ θ2 = 1

3π or 43π


n) : rαi · Dβv ∈ L2(Ωin), |β| = 2



:=‖ v ‖2

1,Ωin

+∑

|β|=2 ‖ rαi · Dβv ‖20,Ωi

n

1/2.



REGULARITY RESULTS

Theorem


n), α1 > 25 ,

u2n ∈ H2,α2(Ω2

n), α2 > 14 , un ∈ C(Ω), un|Kn ∈ H2(Kn).

A B

Ω1n =⇒ θ1 = 2

3π or 53π;

Ω2n =⇒ θ2 = 1

3π or 43π


n) : rαi · Dβv ∈ L2(Ωin), |β| = 2



:=‖ v ‖2

1,Ωin

+∑

|β|=2 ‖ rαi · Dβv ‖20,Ωi

n

1/2.



NUMERICAL APPROXIMATION OF (Pn), n ∈ N fixed:F.E.M + FINITE DIFFERENCES

Ωin are not convex polygons ⇒ ui

n /∈ H2(Ωin) ⇒ ADAPTIVE

REFINEMENT MESHES (E. Vacca 2006),(Cefalo,L.,Vacca 2011).Triangulation of Ω: Tn,h = family of triangles T s.t

Ω = ∪T∈Tn,h

T

of size h := maxhT ; T ∈ Tn,h; hT diameter of T ∈ Tn,h.As ui

n has a singular behaviour in small neighborhoods ofreentrant corners of ∂Ωi

n ⇒ NESTED REFINEMENTSTn,hjj∈N, hj → 0, j → +∞


NESTED REFINEMENTSTn,hjj∈N, hj → 0, j → +∞

all vertices of Kn are nodes of Tn,hj , j ∈ N;Tn,hjj∈N is a regular and not quasi uniform family ofconformal meshes;the variation in size is consistent with “Grisvard’sassumptions":∃ σ > 0 s.t., as h tends to zero, we have(a) hT ≤ σ h1/(1−αi ) for all T ∈ Tn,h such that T ∩ Kn 6= ∅(b) hT ≤ σ h infT rαi for all T ∈ Tn,h such that T ∩ Kn = ∅where α1 = 2/5 + ε, α2 = 1/4 + ε, ε > 0 and r(x) is theweighing distance.


A triangulation Tn,hj of Ωni is conformal if

Ωni = ∪T

T∈Tn,h·T 6= ∅·T 1 ∩

·T 2 = ∅ ∀T1, T2 ∈ Tn,hj : T1 6= T2

T1 ∩ T2 6= ∅, T1 6= T2 =⇒ T1 ∩ T2 = edge or one vertexA family of triangulations Tn,hj is regular if∃σ ≥ 1 : max hT

ρTT∈Tn,hj

≤ σ,∀h > 0,

ρT = supdiam(B) : Bball ⊂ TA family of triangulations Tn,hj is quasi uniform if it isregular and∃τ > 0s.t .min hT

T∈Tn,hj

≥ τh, ∀h > 0


From left to right: T1,h1 Coarse mesh, T1,h2 (global ref.), T1,h3

(local ref.). we want that subsequent applications of the meshalgorithm produce triangulation "conformal with pre-fractals" ofincreasing order j ≥ n


we want that subsequent applications of the mesh algorithmproduce triangulation "conformal with pre-fractals" of increasingorder j ≥ n



SEMIDISCRETE APPROXIMATION OF (Pn), n fixedLet Vn,h(Ω) = v ∈ C0(Ω), v =0 on ∂Ω v |T polynomial of degree 1

Vn,h(Ω) ⊂ V (Ω, Kn), dimVn,h(Ω) = Nh, numb. of inner nodes of Tn,h

Given f in Cδ([0, T ], L2(Ω, m)), for each t ∈ [0, T ],find un,h(t) ∈ Vn,h(Ω) such that ∀ vh ∈ Vn,h(Ω)

(Pn,h)

( d

dt un,h, vh)L2(Ω,m) + En(un,h(t), vh) =∫Ω f vh dm,

un,h(0) = 0.(3)

TheoremTheorem 5 For each t ∈ [0, T ] ∃ a unique un,h(t) ∈ Vn,h(Ω).


ESTIMATE OF THE ORDER OF CONVERGENCE of un,h to un

Theorem

Let un be the solution of (Pn) and un,h(t) the solution of (Pn,h);for each t ∈ [0, T ] there exists c indipendent from h :

‖ un(t)− un,h(t) ‖2L2(Ω,m) +

∫ t

0‖ un(t)− un,h(t) ‖2

V (Ω,Kn)≤

≤ C h2∫ T

0‖ f (τ) ‖2

L2(Ω,m) dt .

(4)

tools:by adapting Grisvard’s results


Theorem(Vacca,2006),(L.,Vacca 2008)Let un be the solution of (Pn), ui

n be the restriction to Ωin of un,

for i = 1, 2 and Ih(un) its interpolant, then, for each t ∈ [0, T ],there exist positive constants c1, c2 independent from h suchthat:

‖un(t)−Ih(un)(t)‖2V (Ω,Kn)

≤ c1h2(∑

i=1,2

‖uin(t)‖2

H2,αi (Ωin)

+‖un(t)‖2H2(Kn)

)

‖un(t)− Ih(un)(t)‖2L2(Ωi

n)≤ c2h4‖un‖2

H2,αi (Ωin)

i = 1, 2


FULLY DISCRETIZATIONConsider a uniform mesh for the time variable t ,define

tl := l4t , l = 0, 1, · · · ,L, 4t > 0, L := [T/4t ].

We construct a sequence uln,h(P) that approximates the exact

solution un(tl , P) by applying the θ-scheme to the semidiscreteapproximation, we obtain the following problem:find ul

n,h ∈ Vn,h(Ω) such that

(P ln,h)

14t (u

l+1n,h − ul

n,h, vh)L2(Ω,m) + E (n)(θul+1n,h + (1− θ)ul

n,h, vh) =

= (θf (tl+1) + (1− θ)f (tl), vh)L2(Ω,m), ∀vh ∈ Vn,h(Ω)

u0n,h = 0,

(5)for each l = 0, 1, · · · ,L.The θ-scheme is unconditionally stable w.r.t. L2(Ω, m)-normprovided that 1

2 ≤ θ ≤ 1. In the case of 0 ≤ θ < 12 , one has to

assume that Tn,hjj∈N is a quasi-uniform family oftriangulations and that a restriction on the time step holds.


FULLY DISCRETIZATIONConsider a uniform mesh for the time variable t ,define

tl := l4t , l = 0, 1, · · · ,L, 4t > 0, L := [T/4t ].

We construct a sequence uln,h(P) that approximates the exact

solution un(tl , P) by applying the θ-scheme to the semidiscreteapproximation, we obtain the following problem:find ul

n,h ∈ Vn,h(Ω) such that

(P ln,h)

14t (u

l+1n,h − ul

n,h, vh)L2(Ω,m) + E (n)(θul+1n,h + (1− θ)ul

n,h, vh) =

= (θf (tl+1) + (1− θ)f (tl), vh)L2(Ω,m), ∀vh ∈ Vn,h(Ω)

u0n,h = 0,

(5)for each l = 0, 1, · · · ,L.The θ-scheme is unconditionally stable w.r.t. L2(Ω, m)-normprovided that 1

2 ≤ θ ≤ 1. In the case of 0 ≤ θ < 12 , one has to

assume that Tn,hjj∈N is a quasi-uniform family oftriangulations and that a restriction on the time step holds.


FULLY DISCRETIZATIONERROR ESTIMATE between the semi-discrete solution un,h(tl)and the fully discrete one ul

n,h for any fixed h.

Theorem

Assume that f ∈ Cδ([0, T ]; L2(Ω, m)) and∂f∂t ∈ L2([0, T ]× Ω, dt × dm), then the functions ul

n,h and un,h(tl)satisfy

‖uln,h − un,h(tl)‖2

L2(Ω,m) ≤

Cθ(4t)2(‖∂un,h

∂t (0)‖2L2(Ω,m)

+∫ T

0 ‖∂f (s)∂t ‖2

L2(Ω,m)

),for

each l=0,1,· · · ,L.


STABILITY RESULT (see e.g. Quarteroni Valli).Corollary 8 Let ul

n,h be the solution of problem (P ln,h), then it

satisfies

‖uln,h‖L2(Ω,m) ≤ Cθ

(sup

t∈[0,T ]

‖f‖L2(Ω,m)

), l = 0, 1, · · · ,L,

where the constant Cθ is a non-decreasing function of thecontinuity constant of E (n)(·, ·) and T and it is also independentof L,4t and h.


CONVERGENCE RESULTS (Cefalo, Dell’Acqua, L. 2011)

Theorem

Let un(t) be the solution of problem (Pn), uln,h be the solution of

problem (P ln,h) and let 1

2 ≤ θ ≤ 1 then it holds

‖un(tl)− uln,h‖2

L2(Ω,m) ≤ ch2∫ T

0‖f (τ)‖2

L2(Ω,m) dτ+

+Cθ4t2 ·(‖∂un,h

∂t(0)‖2

L2(Ω,m) +

∫ T

0‖∂f (τ)

∂t‖2

L2(Ω,m) dτ).

for each l = 0, 1, · · · ,L.


Comments: the layer acts as a preferred escape way for theheat produced by the source term the prefractal layerpenetrates in Ω1

i and it drains heat mainly from Ω1i .

Simulations require a huge amount of computational power andmemory AIM:to carry out simulations on a cluster of computersexploiting parallel programming techniques to computeefficiently the mesh and the solutions over a distributed system

J Stationary solutions using differentmesh,a flat coloring is used to point out where the numericsolution has been computed


Prefractal interface K2“Nested refinement meshes" for the problem

(P2,j)j=3,4,5


Prefractal interface K1Discrete solutions of (P1,j)j=2,3,4












The SNOWFLAKE-type F


THE Koch-type pipe S = F × I, I = [0, 1]


GREEN FORMULAS (L.2002-2003)Let u(t , ·) ∈ C([0, T ]; V (Qi)), where

V (Qi) =

u ∈ H10 (Q); 4ui ∈ L2(Qi)

,

( the Laplacian is intended in the distributional sense).For every fixed t , the normal derivative ∂ui

∂niis in the dual(

(B2,2β,0)(S)

)′of the space

((B2,2

β,0)(S))

, where β = df2 and

⟨∂ui

∂ni, v |S

⟩((B2,2

β,0)(S))′,((B2,2β,0)(S))

=

∫Qi

Du(t , P) Dv(P)dQ+

∫Qi

v(P) 4ui(t , P)dQ

for every t ∈ [0, T ] and every v ∈ H10 (Q).......

∂ui

∂ni∈ C([0, T ]; (B2,2

β,0)(S))′)

.


THE LAGRANGIAN ON KE [u] is a Dirichlet form of diffusion type =⇒ it admits an integralrepresentation :There exists a unique positive Radon measure, which we callLK [u] such that

E [u] =

∫K

dLK [u],

which is uniquely defined by∫K

ϕdLK [u] = E(ϕu, u)− 12

E(ϕ, u2),

∀ϕ ∈ D (E) ∩ C0(K )


TRACES on S (Jonsson-Wallin, Triebel)S is a d–set:∃c1, c2 > 0 : c1rd ≤ m(B(P, r) ∩ S) ≤ c2rd ∀P ∈ S, (Hutchinson ,Falconer).B2,2

β (S) = f ∈ L2(S, m) : ‖f‖B2,2β (S)

< ∞.‖f‖B2,2

β (S):=(

‖f‖2L2(S,m)

+∫∫|P−P′|<1

|f (P)−f (P′)|2|P−P′|2β+d dm(P)dm(P ′)

)1/2

For f in H1(Q), we putγ0f (P) = limr→0

1|B(P,r)∩Q|

∫B(P,r)∩Q f (P ′)dP ′

at every P ∈ Q where the limit exists. The above limit exists atquasi every P ∈ Q with respect to the Newtonian-capacity (seeAdams and Hedberg ).


TRACE THEOREMS,(Jonsson-Wallin, Triebel)Let Q ⊂ R3. Then B2,2

d/2(S) is the trace space to S of H1(Q) inthe following sense:

γ0 is a continuous and linear operator from H1(Q) toB2,2

d/2(S),

there exists a continuous linear operator Ext from B2,2d/2(S)

to H1(Q) such that γ0 Ext is the identity operator inB2,2

d/2(S).

The space B2,2d2 ,0

(S) is a subspace of B2,2d2

(S), more

preciselyB2,2

d2 ,0

(S) = u ∈ L2(S, m)|∃ w ∈ H10 (Q) such that γ0w =

u on S,equipped with the norm‖u‖B2,2

d2 ,0

(S)= inf‖w‖H1(Q) : w ∈ H1

0 (Q), γ0w = u on S.


CONVERGENCE OF HILBERT SPACES (Kuwae-Shyoia 2003)The Hilbert spaces we consider are real and separable.

Definition

A sequence of Hilbert spaces Hhh∈N converges to a Hilbertspace H if there exists a dense subspace C⊂ H and asequence Φhh∈N of linear operatorsΦh : C ⊂ H → Hh such that

limh→∞

‖Φhu‖Hh= ‖u‖H for any u ∈ C

Definition(Strong convergence) A sequence of vectors uhh∈N withuh ∈Hh strongly converges to a vector u∈H if there exists asequence

um

m∈N tending to u in H such that

limm→∞

limh→∞

∥∥Φhum- uh∥∥

Hh= 0


Definition(Weak convergence) A sequence of vectors uhh∈N withuh ∈Hh weakly converges to a vector u∈H if

(uh, vh)Hh→ (u,v)H

for every sequence vhh∈N strongly tending to v ∈H.

Lemma

Let uhh∈N be a sequence with uh ∈Hh,which weaklyconverges to u∈H. Then sup

h→∞‖uh‖Hh

< ∞, ‖u‖H ≤ limh→∞

‖uh‖Hh.

Moreover, uh → u strongly if and only if ‖u‖H = limh ‖uh‖Hh .


characterizations of the strong convergence

Lemma

Let u∈H and let uhh∈N be a sequence of vector uh ∈Hh. Thenuhh∈N strongly converges to u∈H if and only if

(uh, vh)Hh→ (u,v)H

for every sequence vhh∈N with vh ∈ Hh weakly converging toa vector v ∈ H

Lemma

A sequence of vectors uhh∈N with uh ∈Hh strongly convergesto a vector u∈H if and only if

‖uh‖Hh→ ‖u‖H and

(uh, Φh(ϕ))Hh→ (u,ϕ)H for every ϕ ∈ C


Lemma

Let uhh∈N be a sequence with uh ∈Hh. If ‖uh‖Hhis uniformly

bounded for h∈ N, there exists a weakly convergentsubsequence of uhh∈N .


We define the space H = ∪ Hh as the union of Hh and definestrong and weak convergence in H respectively as inDefinitions ?? and ??.

Lemma

For every u ∈ H there exists a sequence uhh∈N, uh ∈Hhstrongly converging to u in H.

Definition

A sequence of bounded operators Bhh∈N, Bh ∈ L(Hh),strongly converges to an operator B ∈ L(H), if for everysequence of vectors uhh∈N with uh ∈ Hh strongly convergingto a vector u ∈ H, the sequence Bhuh strongly converges toBu ∈ H.


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