Corrector estimates for the homogenization of a two-scale thermoelasticity problem with a priori known phase transformations

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Eden, M., Muntean, A. (2017)

Corrector estimates for the homogenization of a two-scale thermoelasticity problem with a priori known phase transformations.

Electronic Journal of Differential Equations, (57): 1-21

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

CORRECTOR ESTIMATES FOR THE HOMOGENIZATION OF A TWO-SCALE THERMOELASTICITY PROBLEM WITH A

PRIORI KNOWN PHASE TRANSFORMATIONS

MICHAEL EDEN, ADRIAN MUNTEAN Communicated by Ralph Showalter

Abstract. We investigate corrector estimates for the solutions of a thermoe- lasticity problem posed in a highly heterogeneous two-phase medium and its corresponding two-scale thermoelasticity model which was derived in [11] by two-scale convergence arguments. The medium in question consists of a con- nected matrix with disconnected, initially periodically distributed inclusions separated by a sharp interface undergoing a priori known phase transforma- tions. While such estimates seem not to be obtainable in the fully coupled set- ting, we show that for some simplified scenarios optimal convergence rates can be proven rigorously. The main technique for the proofs are energy estimates using special reconstructions of two-scale functions and particular operator es- timates for periodic functions with zero average. Here, additional regularity results for the involved functions are necessary.

1. Introduction

We aim to derive quantitative estimates that show the quality of the upscal- ing process of a coupled linear thermoelasticity system with a priori known phase transformations posed in a high-contrast media to its corresponding two-scale ther- moelasticity system.

The problem we have in mind is posed in a medium where the two building components, initially assumed to be periodically distributed, are different solid phases of the same material in which phase transformations that are a priori known occur. As a main effect, the presence of phase transformations leads to evolution problems in time-dependent domains that are not necessarily periodic anymore.

In our earlier paper [11], we rigorously studied the well-posedness of such a ther- moelasticity problem and conducted a homogenization procedure via the two-scale convergence technique (cf. [1] for details). Those results were obtained after trans- forming the problem to a fixed reference geometry. In this work, our goal is to further investigate the connection of those problems and to derive an upper bound for the convergence rate (in some yet to be defined sense) of their solutions. While historically a tool to also justify the homogenization (via asymptotic expansions) in

2010 Mathematics Subject Classification. 35B27, 35B40, 74F05.

Key words and phrases. Homogenization; two-phase thermoelasticity; corrector estimates;

time-dependent domains; distributed microstructures.

c

2017 Texas State University.

Submitted February 12, 2017. Published February 23, 2017.

1

the first place, such estimates, which in the homogenization literature are usually called error and corrector estimates, provide a means to evaluate the accuracy of the upscaled model. Also, such estimates are especially interesting from a compu- tational point of view. In the context of Multiscale FEM, for example, they are needed to ensure/control the convergence of the method, we refer to, e.g., [2, 15].

The basic idea is to estimate the L ² − and H ¹ -errors of the solutions of the problems using energy techniques, additional regularity results, and special operator estimates for functions with zero average. Since, in general, the solutions of the ε-problem do not even have the same domain as the solutions of the two-scale problem, we additionally rely on so-called macroscopic reconstructions (we refer to Section 3). The difficulty in getting such estimates in our specific scenario is twofold: First, the coupling between the quasi-stationary momentum equation and the heat equation and, second, the interface movement which (after transforming to a reference domain) leads to additional terms as well as time-dependent and non-periodic coefficients functions.

As typical for a corrector estimate result, the main goal is to show that there is a constant C > 0 which is independent on the particular choice of ε such that

kΘ ^ε _err k L

^∞

(S×Ω) + kU _err ^ε k L

^∞

(S;L

²

(Ω))

³

+ k∇Θ ^ε _cor k L

²

(S×Ω

^εA

)

³

+ k∇U _cor ^ε k _L

∞

(S;L

²

(Ω

^ε_A

))

^3×3

+ εk∇Θ _err k _L

2

(S×Ω

^εB

)

³

+ εk∇U _cor ^ε k _L

∞

(S;L

²

(Ω

^ε_B

))

^3×3

≤ C( √ ε + ε).

(1.1)

Here, Ω represents the full medium, Ω ^ε _A the fast-heat-conducting connected matrix and Ω ^ε _B the slow-heat-conducting inclusions. For the definitions of the (error and corrector) functions, we refer the reader to the beginning of Section 4.

Unfortunately, in the general setting of a fully-coupled thermoelasticity problem with moving interface, such corrector estimates as stated in (1.1) seem not to be obtainable; in Section 4.3, we point out where and why the usual strategy for establishing such estimates is bound to fail.

Instead, we show that there are a couple of possible simplifications of the full model in which (1.1) holds:

(a) Weakly coupled problem: If we assume either the mechanical dissipation or the thermal stresses to be negligible, we are led to weakly coupled prob- lems, where the desired estimates can be established successively, see The- orem 4.3. We note that the regularity requirements are higher in the case of no thermal stress compared to the case of mechanical dissipation.

(b) Microscale coupling : If mechanical dissipation and thermal stress are only really significant in the slow-conducting component and negligible in the connected matrix part, the estimates hold, see Theorem 4.4.

As pointed out in [25], neglecting the effect of mechanical dissipation is a step that is quite usual in modeling thermoelasticity problems.

Convergence rates for specific one-phase problems with periodic constants (some

of them posed in perforated domains) were investigated in, e.g., [3, 4, 6]. In [10],

convergence rates for a complex nonlinear problem modeling liquid-solid phase tran-

sitions via a phase-field approach were derived. A homogenization result including

corrector estimates for a two-scale diffusion problem posed in a locally-periodic ge-

ometry was proven in [19, 24]. Here, similar to our scenario, the microstructures

are non-uniform, non-periodic, and assumed to be a priori known; the microstruc-

tures are however time independent and there are no coupling effects. For some

corrector estimate results in the context of thermo and elasticity problems, we refer to [3, 22], e.g. We also want to point out to the newer and different philosophy in which the solutions are compared in the two-scale spaces (e.g., L ² (Ω; H _# ¹ (Y B )) as opposed to the, possibly ε-dependent, spaces for the ε-problem (e.g., L ² (Ω ^ε _B )), ¹ a method which requires considerably less regularity on part of the solutions of the homogenized problems, we refer to [14, 18, 21].

The paper is organized as follows: In Section 2, we introduce the ε-microscopic geometry and formulate the thermoelasticity micro-problem in the moving geometry and transformed to a fixed reference domain and also state the homogenized two- scale problem. The assumptions on our data, some regularity statements, and auxiliary estimate results are then collected in Section 3. Finally, in Section 4, we focus on establishing convenient ε-control for the terms arising in the error formulation (see (4.1)). Based on these estimates, the corrector estimate (1.1) is then shown to hold for the above described cases (a) and (b).

2. Setting

2.1. Interface movement. The following notation is taken from [11]. Let S = (0, T ), T > 0, be a time interval. Let Ω be the interior of a union of a finite number of closed cubes Q _j , 1 ≤ j ≤ n, n ∈ N, whose vertices are in Q ³ such that, in addition, Ω is a Lipschitz domain.

In addition, we denote the outer normal vector of Ω with ν = ν(x). Let Y = (0, 1) ³ be the open unit cell in R ³ . Take Y A , Y B ⊂ Y two disjoint open sets, such that Y _A is connected, such that Γ := Y _A ∩ Y B is a C ³ interface, Γ = ∂Y _B , Y _B ⊂ Y , and Y = Y _A ∪ Y _B ∪ Γ. With n = n(y), y ∈ Γ, we denote the normal vector of Γ pointing outwards of Y _B .

Y

B

(0) Y

A

(0)

Γ(0)

Ω

^εB

(0) Ω

^εA

(0)

Γ

^ε

(0) ε

1

Figure 1. Reference geometry and the resulting ε-periodic initial configuration. Note that for t 6= 0, these domains typically loose their periodicity.

For ε > 0, we introduce the εY -periodic, initial domains Ω ^ε _A and Ω ^ε _B and interface Γ ^ε representing the two phases and the phase boundary, respectively, via (i ∈ {A, B})

Ω ^ε _i = Ω ∩ ∪ _k∈Z

3

ε(Y i + k)
, Γ ^ε = Ω ∩ ∪ _k∈Z

3

ε(Γ + k)
.

1 Here, we have used notation and domains as introduced in Section 2.

Here, for a set M ⊂ R ³ , k ∈ Z ³ , and ε > 0, we employ the notation ε(M + k) := x ∈ R ³ : x

ε − k ∈ M .

From now on, we take ε = (ε n ) _n∈N to be a sequence of monotonically decreasing positive numbers converging to zero such that Ω can be represented as the union of cubes of size ε. Note that this is possible due to the assumed structure of Ω.

Here n ^ε = n( ^x _ε ), x ∈ Γ ^ε , denotes the unit normal vector (extended by period- icity) pointing outwards Ω ^ε _B into Ω ^ε _A . The above construction ensures that Ω ^ε _A is connected and that Ω ^ε _B is disconnected. We also have that ∂Ω ^ε _B ∩ ∂Ω = ∅.

We assume that s : S × Ω × R ³ → Y is a function such that (1) s ∈ C ² (S; C ² (Ω) × C _# ² (Y )), ²

(2) s(t, x, ·) _|Y : Y → Y is bijective for every (t, x) ∈ S × Ω, (3) s ^{− 1} ∈ C ² (S; C ² (Ω) × C _# ² (Y )), ³

(4) s(0, x, y) = y for all x ∈ Ω and all y ∈ Y ,

(5) s(t, x, y) = y for all (t, x) ∈ S × Ω and all y ∈ ∂Y ,

(6) there is a constant c > 0 such that dist(∂Y, γ) > c for all γ ∈ s(t, x, Γ) and (t, x) ∈ S × Ω,

(7) s(t, x, y) = y for all (t, x) ∈ S×Ω and for all y ∈ Y such that dist(∂Y, y) < ^c ₂ , (8) there are constants c _s , C _s > 0 such that

c s ≤ det(∇s(t, x, y)) ≤ C s , (t, x, y) ∈ S × Ω × R ³ and set the (t, x)-parametrized sets

Y A (t, x) = s(t, x, Y A ), Y B (t, x) = s(t, x, Y B ), Γ(t, x) = s(t, x, Γ).

We introduce the operations

[·] : R ³ → Z ³ , [x] = k such that x − [x] ∈ Y, {·} : R ³ → Y, {x} = x − [x]

and define the ε-dependent function ⁴

s ^ε : S × Ω → R ³ , s ^ε (t, x) := ε  x ε  + εs 

t, ε  x ε , x

ε

 .

For i ∈ {A, B} and t ∈ S, we set the time dependent sets Ω ^ε _i (t) and Γ ^ε (t) and the corresponding non-cylindrical space-time domains Q ^ε _i and space-time phase boundary Σ ^ε via

Ω ^ε _i (t) = s ^ε (t, Ω ^ε _i ), Q ^ε _i = ∪ t∈S ({t} × Ω ^ε _i (t)) , Γ ^ε (t) = s ^ε (t, Γ ^ε ), Σ ^ε = ∪ _t∈S ({t} × Γ ^ε (t)) ,

and denote by n ^ε = n ^ε (t, x), t ∈ S, x ∈ Γ ^ε (t), the unit normal vector pointing outwards Ω ^ε _B (t) into Ω ^ε _A (t). The time-dependent domains Ω ^ε _i host the phases at time t ∈ S and model the movement of the interface Γ ^ε . We emphasize that, for any t 6= 0, the sets Ω ^ε _A (t), Ω ^ε _B (t), and Γ ^ε (t) are, in general, not periodic.

2 Here, and in the following, the # subscript denotes periodicity, i.e., for k ∈ N

0

, we have C

_#^k

(Y ) = {f ∈ C

^k

(R

³

) : f (x + e

_i

) = f (x) for all x ∈ R

³

}, e

_i

basis vector of R

³

.

3 Here, s

⁻¹

: S × Ω × R

³

→ Y is the unique function such that s(t, x, s

⁻¹

(t, x, y)) = y for all (t, x, y) ∈ S × Ω ∈ Y extended by periodicity to all y ∈ R

³

.

4 This is the typical notation in the context of homogenization via the periodic unfolding method,

see, e.g., [8, 9].

We introduce the transformation-related functions (here, W _Γ ^ε is the normal ve- locity and H _Γ ^ε the mean curvature of the interface) via

F ^ε : S × Ω → R ^3×3 , F ^ε (t, x) := ∇s ^ε (t, x), J ^ε : S × Ω → R, J ^ε (t, x) := det (∇s ^ε (t, x)) ,

v ^ε : S × Ω → R ³ , v ^ε (t, x) := ∂ t s ^ε (t, x), W _Γ ^ε : S × Γ ^ε → R, W _Γ ^ε (t, x) := v ^ε (t, x) · n ^ε (t, s ^ε (t, x)), H _Γ ^ε : S × Γ ^ε → R, H _Γ ^ε (t, x) := − div (F ^ε ) ^{− 1} (t, x)n ^ε (t, s ^ε (t, x))
for which we have the following estimates

kF ^ε k _L

^∞

_(S×Ω)

3×3

+ k(F ^ε ) ^{− 1} k _L

^∞

_(S×Ω)

3×3

+ kJ ^ε k _L

^∞

_(S×Ω)

+ ε ^{− 1} kv ^ε k _L

^∞

_(S×Ω)

3

+ ε ^{− 1} kW _Γ ^ε k _L

^∞

_(S×Γ

ε

) + εkH _Γ ^ε k _L

^∞

_(S×Γ

ε

) ≤ C. (2.1) By design, the constant C entering (2.1) is independent on the choice of ε. Note that the same estimates also hold for the time derivatives of these functions.

2.2. ε-problem and homogenization result. The bulk equations of the coupled thermoelasticity problem are given as (we refer to [5, 11, 16])

− div(C _A ^ε e(u ^ε _A ) − α ^ε _A θ A I 3 ) = f _u ^ε

_A

in Q ^ε _A , (2.2a)

− div(C _B ^ε e(u ^ε _B ) − α ^ε _B θ B I 3 ) = f _u ^ε

_B

in Q ^ε _B , (2.2b)

∂ _t (ρ _A c _dA θ _A ^ε + γ _A ^ε div u ^ε _A ) − div(K _A ^ε ∇θ _A ^ε ) = f _θ ^ε

A

in Q ^ε _A , (2.2c)

∂ t (ρ B c dB θ _B ^ε + γ _B ^ε div u ^ε _B ) − div(K _B ^ε ∇θ ^ε _B ) = f _θ ^ε

_B

in Q ^ε _B . (2.2d) Here, C _i ^ε ∈ R ^3×3×3×3 are the stiffness tensors, α ^ε _i > 0 the thermal expansion coefficients, ρ _i > 0 the mass densities, c _di > 0 the heat capacities, γ _i ^ε > 0 are the dissipation coefficients, K _i ^ε ∈ R ^3×3 the thermal conductivities, and f _u ^ε

i

, f _θ ^ε

i

are volume densities. In addition, e(v) = 1/2(∇v + ∇v ^T ) denotes the linearized strain tensor and I 3 the identity matrix.

At the interface between the phases, we assume continuity of both the tempera- ture and deformation and the fluxes of force and heat densities to be given via the mean curvature and the interface velocity, resp.: ⁵

Ju

ε

K = 0, Jθ

ε

K = 0 on Σ ^ε , (2.2e)

JC

ε ε(u ^ε ) − α ^ε θ ^ε I 3 Kn

ε = −ε ² H _Γ ^ε n ^ε on Σ ^ε , (2.2f) Jρc ^d Kθ

ε W _Γ ^ε + Jγ

ε div u ^ε KW

Γ ε − JK

ε ∇θ ^ε K · n

ε = L AB W _Γ ^ε in Σ ^ε . (2.2g) Here, JvK := v ^A − v B denotes the jump across the boundary separating phase A from phase B, σ ₀ > 0 is the coefficient of surface tension, and L AB ∈ R is the latent heat

Finally, at the boundary of Ω and for the initial condition, we pose

u ^ε _A = 0 on S × ∂Ω ^ε _A , (2.2h)

θ _A ^ε = 0 on S × ∂Ω ^ε _A , (2.2i)

θ ^ε (0) = θ ₀ ^ε on Ω, (2.2j)

u ^ε (0) = 0 on Ω, (2.2k)

5 Here, the scaling via ε

²

counters the effects of both the interface surface area, note that

ε|Γ

^ε

| ∈ O(1), and the curvature itself, note that ε| f H

_Γ^ε

| ∈ O(1).

where θ ^ε ₀ is some (possibly highly heterogeneous) initial temperature distribution.

The scaling of the coefficients is chosen as

C _A ^ε = C A , K _A ^ε = K A , α ^ε _A = α A , γ _A ^ε = γ A , C _B ^ε = ε ² C B , K _B ^ε = ε ² K B , α ^ε _B = εα B , γ ^ε _B = εγ B .

Remark 2.1. The simplified models described in the introduction (for which cor- rector estimates can be established) correspond to α A = α B = 0 or γ A = γ B = 0 (case (1), weakly coupled problem) and α A = γ A = 0 (case (2), mirco coupled problem).

Now, taking the back-transformed quantities (defined on the initial, periodic domains Ω ^ε _i ) U _i ^ε : S × Ω ^ε _i → R ³ and Θ ^ε _i : S × Ω ^ε _i → R ³ given via U _i ^ε (t, x) = u ^ε _i (t, s ^{− 1,ε} (t, x)) and Θ ^ε _i (t, x) = θ _i ^ε (t, s ^{− 1,ε} (t, x)), ⁶ we obtain the following prob- lem in fixed coordinates (for more details regarding the transformation to a fixed domain, we refer to [9, 17, 20]):

⁷

− div (C _A ^r,ε (U _A ^ε ) − Θ ^ε _A α ^r,ε _A ) = f _u ^r,ε

_A

in S × Ω ^ε _A , (2.3a)

− div ε ² C _B ^r,ε e(U _B ^ε ) − εΘ ^ε _B α ^r,ε _B
= f _u ^r,ε

_B

in S × Ω ^ε _B , (2.3b)

∂ t (c ^r,ε _A Θ ^ε _A + γ _A ^r,ε : ∇U _A ^ε ) − div (K _A ^r,ε ∇Θ ^ε _A )

− div ((c ^r,ε _A Θ ^ε _A + γ ^r,ε _A : ∇U _A ^ε ) v ^r,ε )

= f _θ ^r,ε

A

in S × Ω ^ε _A ,

(2.3c)

∂ _t (c ^r,ε _B Θ ^ε _B + εγ _B ^r,ε : ∇U _B ^ε ) − div ε ² K _B ^r,ε ∇Θ ^ε _B

− div ((c ^r,ε _B Θ ^ε _B + εγ _B ^r,ε : ∇U _B ^ε ) v ^r,ε )

= f _θ ^r,ε

B

in S × Ω ^ε _B ,

(2.3d)

complemented with interface transmission, boundary, and initial conditions.

Now, for j, k ∈ {1, 2, 3} and y ∈ Y , set d _jk = (y _j δ _1k , y _j δ _2k , y _j δ _3k ) ^T . For t ∈ S, x ∈ Ω, let τ _j ^θ (t, x, ·) ∈ H ¹

# (Y A ), τ _jk ^u (t, x, ·), τ ^u (t, x, ·) ∈ H ¹

# (Y A ) ³ be the solutions to the following variational cell problems ⁸

0 = Z

Y

_A

K _A ^r ∇ y τ _j ^θ + e j
· ∇ y v dy for all v ∈ H _# ¹ (Y A ), (2.4a) 0 =

Z

Y

A

C _A ^r e y (τ _jk ^u + d jk ) : e y (v) dy for all v ∈ H _# ¹ (Y A ) ³ , (2.4b)

0 = Z

Y

_A

C _A ^r e y (τ ^u ) : e y (v A ) dy − Z

Y

_A

α ^r _A : ∇ y v dy for all v ∈ H _# ¹ (Y A ) ³ . (2.4c) Using these functions, we introduce the fourth rank tensor C and the matrix K via

(C) _i

1

i

2

j

1

j

2

= C _A ^r e _y τ _i ^u

1

i

₂

+ d _i

₁

_i

₂

: e y τ _j ^u

1

j

₂

+ d _j

₁

_j

₂

, (K) _ij = K _A ^r ∇ Y τ _j ^θ + e j
· ∇ Y τ _i ^θ + e i
.

6 Here, s

^−1,ε

: S × Ω → Ω is the inverse function of s

^ε

.

7

Here, the superscript r, ε denotes the transformed quantities (w.r.t s

^ε

), for example K

^r,ε_A

= J

^ε

(F

^ε

)

⁻¹

K

_A

(F

^ε

)

^−T

(cf. [11]).

8 Here, and in the following, the superscript r denotes the transformed quantities (w.r.t s), e.g.,

K

_A^r

= J (F )

⁻¹

K

_A

(F )

^−T

.

Furthermore, we define the following set of averaged coefficients C ^h =

Z

Y

_A

C dy, α ^h _A =

Z

Y

_A

(α ^r _A C _A ^r e y (τ ^m )) dy, H _Γ ^h =

Z

Γ

H _Γ n ds, f _u ^h =

Z

Y

A

f _u ^r

_A

dy + Z

Y

B

f _u ^r

_B

dy, c ^h = ρ _A c _dA |Y A | +

Z

Y

_A

γ _A ^r : ∇ _y τ ^u dy, W _Γ ^h = Z

Γ

W _Γ ^r ds, K _A ^h =

Z

Y

A

K dy, f _θ ^h =

Z

Y

A

f _θ ^r

_A

dy + Z

Y

B

f _θ ^r

_B

dy, A ^h (Θ _B , U _B ) =

Z

Y

_B

(c ^r _B Θ _B + γ _B ^r U _B ) dy, γ _A ^h = Z

Y

_A

γ _A ^r + γ _A ^r ∇ y τ _jk ^m
dy.

After a homogenization procedure (the details of which are presented in [11]), we obtain the following upscaled two-scale model

− div C _A ^h e(u _A ) − α ^h _A θ _A
= f _u ^h + H _Γ ^h in S × Ω, (2.5a)

∂ t c ^h θ A + γ _A ^h : ∇u A + A ^h (Θ B , U B )
− div K _A ^h ∇θ A
= f _θ ^h − W _Γ ^h

in S × Ω, (2.5b)

− div y (C _B ^r e y (U B ) − α ^r _B Θ B ) = f _u ^r

_B

in S × Ω × Y B , (2.5c)

∂ t (c ^r _B Θ B + γ _B ^r : ∇ y U B ) − div y (K _B ^r ∇ Y Θ B )

− div _y ((c ^r _B Θ ^r _B + γ _B ^r : ∇ _y U _B ) v ^r )

= f _θ ^r

_B

in S × Ω × Y B ,

(2.5d)

U _B = u _A , Θ _B = θ _A on S × Ω × Γ, (2.5e) again, complemented with corresponding initial and boundary values.

3. Preliminaries

In this section, we lay the groundwork for the corrector estimations that are done in Section 4 in stating the existence and regularity results for the solutions and also giving some auxiliary estimates. We introduce the spaces

H ¹ (Ω ^ε _A ; ∂Ω) = u ∈ H ¹ (Ω ^ε _A ) : u = 0 on ∂Ω and, for a Banach space X,

W(S; X) = u ∈ L ² (S; X) : such that ∂ t u ∈ L ² (S; X ⁰ ) .

In general, we will not differentiate (in the notation) between a function defined on Ω and its restriction to Ω ^ε _A or Ω ^ε _B or between a function defined on one of those subdomains and its trivial extension to the whole of Ω. Here, and in the following, C, C ₁ , C ₂ denote generic constants which are independent of ε but whose values might change even from line to line.

For a function f = f (x, y), we introduce the so called macroscopic reconstruction

[f ] ε = [f ] ε (x) = f (x, x/ε). Note that, for general f ∈ L ^∞ (Ω; H ¹ (Y ), [f ] ε may not

even be measurable (see [1]); continuity in one variable, e.g., f ∈ L ^∞ (Ω; C _# (Y )) is

sufficient, though. Applying the chain rule leads to D[f ] _ε = [D _x f ] _ε + 1/ε[D _y f ] _ε ,

where D = ∇, e(·), div(·), for sufficiently smooth functions f .

Assumption (A1). We assume that θ ^ε ₀ ∈ H ¹ (Ω), f _u ^r,ε

_i

∈ C ¹ (S; L ² (Ω ^ε _i )), f _θ ^r,ε

i

∈ L ² (S×Ω ^ε _i ). Furthermore, let θ A

₀

∈ H ¹ (Ω), f _u ^h

_A

∈ C ¹ (S; L ² (Ω)), f _θ ^h

A

∈ L ² (S×Ω) for the macroscopic homogenized part and θ _B

₀

∈ C(Ω; H ¹ (Y _B )), f _u ^r

B

∈ C ¹ (S; L ² (Ω)), f _θ ^r

B

∈ L ² (S × Ω) for the two-scale part. Moreover, we expect the following conver- gence rates to hold for our data

k1 Ω

^ε_A

θ ^ε ₀ − θ A

₀

k L

²

(Ω

^εA

) ≤ C √ ε, k1 Ω

^εB

θ ^ε ₀ − θ B

0



ε k L

²

(Ω

^εB

) + kf _u ^r,ε

B

− f _u ^r

_B



ε k L

²

(Ω

^εB

)

³

+ kf _θ ^r,ε

B

− f _θ ^r

_B



ε k L

²

(Ω

^εB

) ≤ Cε and, in addition, to have the following estimates

Z

Ω

^εA

 (f _u ^r,ε

_A

− f _u ^h

_A

)ϕ(x) 

 dx ≤ Cεkϕk H

¹

(Ω

^εA

) for all ϕ ∈ H ¹ (Ω ^ε _A ; ∂Ω) ³ , Z

Ω

^εA

 (f _θ ^r,ε

A

− f _θ ^h

_A

)ϕ(x) 

 dx ≤ Cεkϕk _H

1

(Ω

^ε_A

) for all ϕ ∈ H ¹ (Ω ^ε _A ; ∂Ω).

If we are also interested in developing estimates for the time derivatives, we need stronger regularity assumptions.

Assumption (A2). Additionally to Assumptions (A1), we also expect the follow- ing convergence rates to hold:

k1 Ω

^ε_A

θ ^ε ₀ − θ A

₀

k _H

1

(Ω

^εA

) ≤ C √ ε, k1 Ω

^ε_B

θ ₀ ^ε − θ B

₀



ε k _H

1

(Ω

^εB

) + k∂ t (f _u ^r,ε

_B

− f _u ^r

_B



ε )k _L

2

(Ω

^εB

)

³

+ k∂ t (f _θ ^r,ε

B

− f _θ ^r

_B



ε )k _L

2

(Ω

^εB

)

≤ Cε.

Moreover, we assume Z

Ω

^εA

|∂ t (f _u ^r,ε

_A

− f _u ^h

_A

)ϕ(x)| dx ≤ Cεkϕk _H

1

(Ω

^ε_A

) for all ϕ ∈ H ¹ (Ω ^ε _A ; ∂Ω) ³ , Z

Ω

^εA

|∂ t (f _θ ^r,ε

A

− f _θ ^h

_A

)ϕ(x)| dx ≤ Cεkϕk H

¹

(Ω

^εA

) for all ϕ ∈ H ¹ (Ω ^ε _A ; ∂Ω).

3.1. Regularity results. To be able to justify the steps in the estimates in Sec- tion 4, some of the involved functions need to be of higher regularity than is guar- anteed via the standard H ¹ -theory for elliptic/parabolic problems. In the following lemmas, we will collect the appropriate regularity results.

Lemma 3.1 (Regularity of cell problem solutions). The solutions of the problems (2.4a)-(2.4c) possess the regularity τ _j ^θ ∈ C ² (S; C ¹ (Ω; W ^2,p (Y _A ))) and τ _jk ^u , τ ^u ∈ C ² (S; C ¹ (Ω; W ^2,p (Y _A ) ³ )) for some p > 3.

Proof. The regularity w.r.t. y ∈ Y A can be derived from standard elliptic regularity theory (we refer to [13] for the general results and [10] for the application to our case of the cell problems). The rest is a direct consequence of the regularity (w.r.t. t ∈ S

and x ∈ Ω) of the involved coefficients. 

Note that this implies, in particular, that the cell problem functions and their gradients (w.r.t. y ∈ Y ) are bounded and that their macroscopic reconstructions are well-defined measurable functions.

In the following, we denote U ^ε = (U _A ^ε , U _B ^ε ) and Θ ^ε = (Θ ^ε _A , Θ ^ε _B ).

Lemma 3.2 (Existence and regularity theorem for the ε-problem). There is a unique (U ^ε , Θ ^ε ) ∈ W(S; H ₀ ¹ (Ω) ³ × H ₀ ¹ (Ω)) solving the variational system (2.3) for which standard energy estimates hold independently of the parameter ε. Further- more, this solution possesses the regularity (U ^ε , Θ ^ε ) ∈ C ¹ (S; H ² (Ω ^ε _A ) ³ ×H ² (Ω ^ε _B ) ³ )×

L ² (S; H ² (Ω ^ε _A ) × H ² (Ω ^ε _B )) such that ∂ _t Θ ^ε ∈ L ² (S; H ¹ (Ω)).

Proof. The proof of the existence of a unique solution and of the energy estimates is given in [11, Theorem 3.7, Theorem 3.8]. As a linear transmission problem (with sufficiently regular coefficients), regularity results apply (we refer to, e.g., [12]; see,

also, [23] for a similar coupling problem). 

Since s ^ε is a diffeomorphism, this leads to a unique solution to the moving interface problem, also. However, while the solution has H ² -regularity, its second derivatives are not necessarily bounded (and in general will not be) independently of ε > 0.

Lemma 3.3 (Existence and regularity theorem for the homogenized problem).

There is a unique

(u A , θ A , U B , Θ B ) ∈ W(S; H ₀ ¹ (Ω) ³ × H ₀ ¹ (Ω) × L ² (Ω; W ^1,2 (Y B ) ³ ) × L ² (Ω; W ^1,2 (Y B ))) solving the variational system (2.5). Furthermore,

(u A , θ A ) ∈ C ¹ (S; H ² (Ω) ³ ) × L ² (S; H ² (Ω)) such that ∂ _t θ _A ∈ L ² (S; H ¹ (Ω)) and

(U B , Θ B ) ∈ C ¹ (S; H ² (Ω; H ² (Y B ) ³ )) × L ² (S; H ² (Ω; H ² (Y B ))) such that ∂ t Θ B ∈ L ² (S; H ² (Ω; H ¹ (Y B ))).

Proof. Existence of a solution is given via the two-scale homogenization procedure outlined in [11] and uniqueness for this linear coupled transmission problem can then be shown using energy estimates. As to the higher regularity, this follows, again, via the regularity of domain, coefficients, and data, we refer to results outlined in

[12, 23, 24]. 

3.2. Auxiliary estimates. For the transformation related quantities, we have the following estimates available as stated in Lemma 3.4 and Lemma 3.5.

Lemma 3.4. There is a constant C > 0 independent of the parameter ε such that kF ^ε − [F ] ε k _L

∞

(S×Ω)

^3×3

+ kJ ^ε − [J ] ε k _L

^∞

_(S×Ω) + kv ^ε − ε[v] ε k _L

^∞

_(S×Ω)

+ kW _Γ ^ε − εW _Γ 

ε k _L

^∞

_(S×Γ) + kH _Γ ^ε − H _Γ 

ε k _L

^∞

_(S×Γ) ≤ Cε The same estimates hold for the time derivatives of those functions.

Proof. We show this, by way of example, only for F ^ε , the other estimates follow in the same way:

kF ^ε − [f ] ε k L

^∞

(S×Ω)

^3×3

= ess sup _{(t,x)∈S×Ω} 

∇ y s t, ε  x ε , x

ε
− ∇ y s t, x, x ε

  ≤ L ε

√ 2 .

Here, L is the Lipschitz constant of F ^ε w.r.t. x ∈ Ω (uniform in S × Y ). 

Based on these estimates and due to the fact that all material parameters are

assumed to be constant in the moving geometry, we obtain the same estimates for

the material parameters (K ^r,e , α ^r,ε and so on) in the reference configuration.

The following lemma is concerned with ε-independent estimates for the macro- scopic reconstruction of periodic functions with zero average. There are several different but similar theorems that can be found in the literature regarding correc- tor estimates in the context of homogenization, we refer to, e.g., [7, 6, 10, 19], but for our purposes the following version suffices:

Lemma 3.5. Let f ∈ L ² (S × Ω ^ε _A ; C _# (Y ) such that Z

Y

_A

f (t, x, y) dy = 0 a.e. in S × Ω ^ε _A . Then, there is a constant c > 0 such that, independently of ε,

Z

Ω

^εA

|[f ] ε (t, x)ϕ(x)| dx ≤ Cεkϕk _H

1

(Ω

^εA

) . for all ϕ ∈ H ¹ (Ω ^ε _A ; ∂Ω).

The above lemma can be proven similarly to the corresponding statements in [6, Lemma 3] and [19, Lemma 5.2].

4. Corrector estimates

In this section, we are concerned with the actual corrector estimates. Recon- structing micro-solutions from the homogenized functions via the [·] ε -operation and subtracting the heat equations (2.2a), (2.2b), (2.5b), and (2.5d) and momentum equations (2.2a), (2.2a), (2.5a), and (2.5c) we obtain

∂ t c ^r,ε _A Θ ^ε _A − κ A c ^h θ A
+ ∂ t γ _A ^r,ε : ∇U _A ^ε − κ A γ _A ^h : ∇u A

− div K _A ^r,ε ∇Θ ^ε _A − κ A K _A ^h ∇θ A
− κ A W _Γ ^h

= f _θ ^r,ε

A

− κ A f _θ ^h ,

(4.1a)

− div C _A ^r,ε (U _A ^ε ) − κ _A C _A ^h e(u _A ) − α ^r,ε _A Θ ^ε _A + κ _A α ^h _A θ _A
+ H _Γ ^h

= f _u ^r,ε

_A

− κ A f _u ^h , (4.1b)

∂ t c ^r,ε _B Θ ^ε _B − c ^r _B Θ B



ε
+ ∂ t εγ _B ^r,ε : ∇U _B ^ε − γ _B ^r : ∇ Y U B



ε

− div ε ² K _B ^r,ε ∇Θ ^ε _B
+  div Y (K _B ^r ∇ Y Θ B ) 

ε

= f _θ ^r,ε

B

− f _θ ^r

_B



ε ,

(4.1c)

− div ε ² C _B ^r,ε e(U _B ^ε ) − εα ^r,ε _B Θ ^ε _B
+ h

div Y (C _B ^r e y (U B ) − α ^r _B Θ B I 3 ) i

ε = f _u ^r,ε

_B

− f _u ^r

_B



ε . (4.1d)

These equations hold in Ω ^ε _A and Ω ^ε _B , respectively. Using the interface and boundary conditions for both the ε-problem and the homogenized problem and then doing an integration by parts, these equations correspond to a variational problem in H ^{− 1} (Ω).

Our strategy in establishing the estimates is as follows: After introducing error and corrector functions and doing some further preliminary estimates, we first, in Section 4.1, concentrate on the momentum part, i.e., equations (4.1b) and (4.1d).

Here, we take the different terms arising in the weak formulation and estimate them individually using the results from Section 3 and usual energy estimation techniques.

Combining those estimates, it is shown that the mechanical error can be controlled

by the heat error, see Remark 4.1. Then, in Section 4.2, we basically do the same for the heat conduction part, i.e., equations (4.1a) and (4.1c), thereby arriving at the corresponding result that the heat error is controlled by the mechanical error (and the time derivative of the mechanical error ), see Remark 4.2.

Finally, in Section 4.3, we go about combining those individual estimates. Here, we show that for the scenarios (a) (Theorem 4.3) and (b) (Theorem 4.4) (as de- scribed in the introduction), we obtain the desired estimates, i.e., equation (1.1).

Moreover, we point out why the same strategy will not work for the full problem.

Now, we introduce the functions the subscripts “err” and “cor” for error and corrector, respectively.

U _err ^ε =

( U _A ^ε − u A in S × Ω ^ε _A

U _B ^ε − [U _B ] _ε in S × Ω ^ε _B , Θ ^ε _err =

( Θ ^ε _A − θ A in S × Ω ^ε _A Θ ^ε _B − [Θ _B ] _ε in S × Ω ^ε _B , U _cor ^ε =

( U _err ^ε − ε  U e 

ε in S × Ω ^ε _A

U _err ^ε in S × Ω ^ε _B , Θ ^ε _cor =

( Θ ^ε _err − ε  Θ e 

ε in S × Ω ^ε _A Θ ^ε _err in S × Ω ^ε _B . The functions e U and e θ are exactly the functions arising in the two-scale limits of the gradients of U _A ^ε and Θ ^ε _A , respectively, and are given as (cf. [11])

U = e X 3 j,k =1

τ _jk ^u e(u A ) jk + τ ^u θ A , Θ = e X 3 j =1

τ ^θ ∂ j θ A .

Because of the corrector part (namely, ε  Θ e 

ε and ε  U e 

ε , respectively), our cor- rector functions will, in general, not vanish at ∂Ω and are therefore not valid choices of test functions for a weak variational formulation of the system given via equations (4.1a)-(4.1d). Because of that, we introduce a smooth cut-off func- tion m ^ε : Ω → [0, 1] such that m ^ε (x) = 0 for all x ∈ Ω ^ε _A such that dist(x, ∂Ω) ≤ εc/2 and m ^ε (x) = 1 for all x ∈ Ω ^ε _A such that dist(x, ∂Ω) ≥ εc. Furthermore, we require the estimate

√ εk∇m ^ε k _L

2

(Ω) +

√

ε ³ k∆m ^ε k _L

2

(Ω) + 1

√ ε k1 − m ^ε k _L

2

(Ω) ≤ C (4.2) to hold independently of the parameter ε. For this cut-off function m ^ε , we set

U _cor0 ^ε =

( U _cor ^ε + (1 − m ^ε )ε  U e 

ε in S × Ω ^ε _A U _cor ^ε − ε 

U e 

ε in S × Ω ^ε _B , Θ ^ε _cor0 =

( Θ ^ε _cor + (1 − m ^ε )ε  Θ e 

ε in S × Ω ^ε _A Θ ^ε _cor − ε 

Θ e 

ε in S × Ω ^ε _B .

Obviously, we then have Θ ^ε _cor0 ∈ H ₀ ¹ (Ω) and U _cor0 ^ε ∈ H ₀ ¹ (Ω) ³ . Owing to the reg- ularity of e U and e Θ (Lemma 3.1) and the estimate (4.2) for m ^ε , these modified correctors admit the following ε-independent estimates (for i = A, B) ⁹

kU _err ^ε k _L

2

(Ω

^ε_i

)

³

≤ kU _cor0 ^ε k _L

2

(Ω

^ε_i

)

³

+ Cε ≤ kU _err ^ε k _L

2

(Ω

^ε_i

)

³

+ 2Cε, ke(U _cor ^ε )k _L

2

(Ω

^ε_A

)

^3×3

≤ ke(U _cor0 ^ε )k _L

2

(Ω

^ε_A

)

^3×3

+ C( √

ε + ε)

≤ ke(U _cor ^ε )k _L

2

(Ω

^ε_A

)

^3×3

+ 2C( √ ε + ε),

9 The same estimates hold when replacing the linearized strain tensor with the gradient

operator.

εke(U _cor ^ε )k _L

2

(Ω

^εB

)

^3×3

≤ εke(U _cor0 ^ε )k _L

2

(Ω

^εB

)

^3×3

+ C( √ ε + ε)

≤ εke(U _cor ^ε )k _L

2

(Ω

^εB

)

^3×3

+ 2C( √ ε + ε), kΘ ^ε _err k _L

2

(Ω

^εi

) ≤ kΘ ^ε _cor0 k _L

2

(Ω

^εi

) + Cε ≤ kΘ ^ε _err k _L

2

(Ω

^εi

) + 2Cε,

k∇Θ ^ε _cor k _L

2

(Ω

^εA

)

³

≤ k∇Θ ^ε _cor0 k _L

2

(Ω

^εA

)

³

+ C( √ ε + ε)

≤ k∇Θ ^ε _cor k L

²

(Ω

^εA

)

³

+ 2C( √ ε + ε), εk∇Θ ^ε _cor k L

²

(Ω

^εB

)

³

≤ εk∇Θ ^ε _cor0 k L

²

(Ω

^εB

)

³

+ C( √

ε + ε)

≤ εk∇Θ ^ε _cor k L

²

(Ω

^εB

)

³

+ 2C( √ ε + ε).

Applying Korns inequality to U _cor0 (see [11, Lemma 3.1]) and using the above estimates, we then get

kU _err ^ε k L

²

(Ω) + k∇U _cor ^ε k L

²

(Ω

^εA

)

³

+ εk∇U _cor ^ε k L

²

(Ω

^εB

)

^3×3

≤ ke(U _cor ^ε )k L

²

(Ω

^εA

)

³

+ εke(U _cor ^ε )k _L

2

(Ω

^εB

)

^3×3

+ C( √

ε + ε). (4.3) 4.1. Estimates for the momentum equations. Let us first concentrate on the mechanical part of the corrector equations, namely equations (4.1b) and (4.1d). Af- ter integrating over Ω, multiplying with a test function ϕ ∈ H ₀ ¹ (Ω) ³ , and integrating by parts, we obtain

Z

Ω

^εA

C _A ^r,ε e(U _A ^ε ) − κ A C _A ^h e(u A )
: e(ϕ) dx − Z

Γ

^ε

κ A C _A ^h e(u A )n ^ε · ϕ ds

| {z }

=:I

^ε1

(t,ϕ)

+ ε ² Z

Ω

^εB

C _B ^r,ε e(U _B ^ε ) − C _B ^r 

ε e([U B ] ε )
: e(ϕ) dx

| {z }

=:I

2^ε

(t,ϕ)

− Z

Ω

^ε_A

α ^r,ε _A Θ ^ε _A − κ _A α ^h _A θ _A
: ∇ϕ dx + Z

Γ

^ε

κ _A α ^h _A θ _A n ^ε · ϕ ds

| {z }

=:I

3^ε

(t,ϕ)

+ ε Z

Ω

^εB

α ^r,ε _B Θ ^ε _B − α ^r _B Θ B



ε
: ∇ϕ dx

| {z }

=:I

4^ε

(t,ϕ)

= Z

Ω

^ε_A

 f _u ^r,ε

A

− κ _A Z

Y

_A

f _u ^r

A

dy



· ϕ dx

| {z }

=:I

5^ε

(t,ϕ)

+ ε ² Z

Γ

^ε

[C _B ^r ] ε e([U B ] ε )n ^ε · ϕ ds − Z

Ω

^εA

κ A

Z

Y

_B

f _u ^r

_B

dy · ϕ dx

| {z }

=:I

6^ε

(t,ϕ)

+ ε ² Z

Γ

^ε

H _Γ ^r,ε n ^ε · ϕ ds − Z

Ω

^εA

κ A H _Γ ^h · ϕ dx

| {z }

=:I

^ε7

(t,ϕ)

+ Z

Ω

^εB

f _u ^r,ε

_B

− f _u ^r

_B



ε
· ϕ dx

| {z }

=:I

8^ε

(t,ϕ)

+I ₉ ^ε (t, ϕ),

where

I ₉ ^ε (t, ϕ)

:=

Z

Ω

^εB

ε ² div ([C _B ^r e x (U B )] ε ) + ε div x (C _B ^r e y (U B )) 

ε + ε div x (α ^r _B Θ B ) 

ε
· ϕ dx.

We now go on estimating these terms individually and then combine the resulting estimates. We proceed for both U _cor0 ^ε and ∂ t U _cor0 as test function choices. While the first choice is the natural one for energy estimates, the second choice is needed in order to merge those estimates with the heat equation estimates.

Taking a look at I ₁ ^ε , we calculate I ₁ ^ε (t, ϕ) =

Z

Ω

^εA

C _A ^r,ε e(U _cor ^ε ) : e(ϕ) dx

+ Z

Ω

^εA

C _A ^r,ε e(U _A ^ε − U _cor ^ε ) − κ A C _A ^h e(u A )
: e(ϕ) dx

− Z

Γ

^ε

κ A C _A ^h e(u A )n ^ε · ϕ ds We also see (via the definition of C and e U )

C _A ^r,ε e(U _A ^ε − U _cor ^ε ) = [C] _ε e(u _A ) + C _A ^r e _y (τ ^u ) 

ε θ _A + C _A ^r,ε − C _A ^r 

ε



e(u A ) + e y ( e U ) 

ε



+ εC _A ^r,ε e x

 U e  

ε

| {z }

R

^ε₁

,

where, using Lemma 3.5, the estimate 

 Z

Ω

^ε_A

R ^ε ₁ ϕ dx 

 ≤ Cεkpk H

¹

(Ω

^εA

)

holds. Now, seeing that R

Γ Ce(u A )n ds = 0 and div y (Ce(u A )) = 0 a.e. in S × Ω and using Lemma 3.5, we obtain

  Z

Ω

^εA

[C] ε − κ A (0)C _A ^h
e(U A ) : e(ϕ) dx − Z

Γ

^ε

κ A (0)Ce(u A )n ^ε · ϕ dx  

=   Z

Ω

^εA

div [C] _ε − κ A (0)C _A ^h
e(U A )
· ϕ dx  

≤ Cεkϕk _H

1

(Ω

^εA

)

for all ϕ ∈ H ¹ (Ω ^ε _A ; ∂Ω). Also, Z

Ω

^εA

C _A ^r,ε e(U _cor ^ε ) : e(U _cor0 ^ε ) dx

≥ C ₁ ke(U _cor ^ε )k ² _L

2

(Ω

^εA

)

^3×3

+ ε Z

Ω

^ε_A

C _A ^r,ε e(U _cor ^ε ) : e (1 − m ^ε )  U e 

ε
dx

≥ C ₁

2 ke(U _cor ^ε )k ² _L

2

(Ω

^ε_A

)

^3×3

− C ₂ (ε + ε ² ).

Here, the second inequality can be shown using the assumptions on m ^ε and the known estimates of the involved functions. In summary, for I ₁ ^ε , we arrive at

I ₁ ^ε (t, U _cor0 ^ε ) − Z

Ω

^εA

C _A ^r e y (τ ^u ) 

ε θ A : e(U _cor0 ^ε ) dx

≥ C ₁ ke(U _cor ^ε )k ² _L

2

(Ω

^ε_A

)

^3×3

− C ₂ (ε + ε ² ).

(4.4)

Now, going on with I ₂ ^ε , it is easy to see that I ₂ ^ε (t, U _cor0 ^ε ) = ε ²

Z

Ω

^ε_B

C _B ^r 

ε (e(U _B ^ε ) − e([U _B ] _ε )) : e(U _cor ) dx + ε ²

Z

Ω

^εB

C _B ^r,ε − C _B ^r 

ε
e(U _B ^ε ) : e(U _cor ) dx

≥ C ₁ ε ² ke(U _cor ^ε )k ² _L

2

(Ω

^εA

)

^3×3

− C ₂ ε ² .

(4.5)

Going forward with term I ₃ ^ε , we decompose α ^r,ε _A Θ ^ε _A − κ A α ^h _A θ _A = α ^r,ε _A − α ^r _A 

ε
Θ ^ε _A + α ^r _A 

ε Θ ^ε _err + α ^r _A 

ε − κ A α ^h _A
θ A (4.6) from which we can estimate the first two terms as

  Z

Ω

^εA

α ^r,ε _A − α ^r _A 

ε
Θ ^ε _A + α ^r _A 

ε Θ ^ε _err
: ∇ϕ   ≤ C 

ε + kΘ ^ε _err k L

²

(Ω

^εA

)

 k∇ϕk L

²

(Ω

^εA

) .

For the remaining term of equation (4.6) combined with the interface integral part of I ₃ ^ε , we obtain

Z

Ω

^ε_A

α ^r _A 

ε − κ A α ^h _A
θ A : ∇ϕ dx + Z

Γ

^ε

κ _A α ^h _A θ _A n ^ε · ϕ ds

= − Z

Ω

^εA

h div x



α ^r _A − κ A

Z

Y

A

α _A ^r dy  θ A

i

ε · ϕ dx

− Z

Ω

^ε_A

κ _A div  Z

Y

_A

C _A ^r e _y (τ ^u ) dyθ _A 

· ϕ dx

− Z

Γ

^ε

α ^r _A 

ε θ A n ^ε · ϕ ds − 1 ε Z

Ω

^εA

h

div y (α ^r _A θ A ) i

ε · ϕ dx We apply Lemma 3.5 to

f ₁ = div x



α ^r _A − κ A

Z

Y

A

α ^r _A dy  θ A

 , f ₂ = div _x 

C ^r _A e _y (τ ^u )) − κ _A Z

Y

_A

C _A ^r e _y (τ ^u ) dy  θ _A 

and recall that τ ^u is a solution of the cell problem (2.4c) (because of this the

“ ¹ _ε  div y



ε ”-terms vanish) which leads to  I ₃ ^ε (t, ϕ) +

Z

Ω

^ε_A

C ^r _A e _y (τ ^u ) 

ε θ _A : e(ϕ) dx   ≤ C 

ε + kΘ ^ε _err k _L

2

(Ω

^ε_A

)

 kϕk _H

1

(Ω

^ε_A

) . (4.7)

Similarly as with I ₂ ^ε , for the thermo-elasticity term I ₄ ^ε , we obtain

|I ₄ ^ε (t, ϕ)| ≤ CkΘ ^ε _err k ² _L

2

(Ω

^ε_B

) + ε ² k∇ϕk ² _L

2

(Ω

^ε_B

)

^3×3

. (4.8) Next, we take the mean curvature error term I ₇ ^ε

|I ₇ ^ε (t, ϕ)| ≤ ε ²   Z

Γ

^ε

H _Γ ^r,ε − H _Γ ^r 

ε
n ^ε ·ϕ ds  + 

ε ² Z

Γ

^ε

H _Γ ^r 

ε n ^ε ·ϕ ds−

Z

Ω

^ε_A

κ _A H _Γ ^h ·ϕ dx 

.

Now, in view of the assumptions on our data and for the source density errors

(stated in Assumptions (A1)) and the curvature estimate, using Lemma 3.4 (for

the functional I ₆ ^ε (t, ϕ)), and the boundedness of the functions involved in I ₉ ^ε , we estimate

|I ₅ ^ε (t, ϕ)| + |I ₆ ^ε (t, ϕ)| + |I ₇ ^ε (t, ϕ)| + |I ₈ ^ε (t, ϕ)| + |I ₉ ^ε (t, ϕ)|

≤ Cε 

kϕk _H

1

(Ω

^εA

) + kϕk _L

2

(Ω

^εB



. (4.9)

Finally, merging the individual estimates for the error terms (namely, estimates (4.4), (4.5), (4.7), (4.8), (4.9)) and the estimate (4.3), we conclude

kU _err ^ε k _L

2

(Ω)

³

+ k∇U _cor ^ε k _L

2

(Ω

^εA

)

^3×3

+ εk∇U _cor ^ε k _L

2

(Ω

^εA

)

^3×3

≤ C √

ε + ε + kΘ _err k L

²

(Ω)

 . (4.10)

Remark 4.1. With estimate (4.10) it is clear that the error in the mechanical part inherits the convergence rate from the heat-error (at least if it is not faster than

√ ε).

If, additionally, Assumptions (A2) are fulfilled, it is also possible to first differ- entiate equations (4.1b) and (4.1d) with respect to time and to then choose the test function ∂ t U _cor for the variational formulation of the arising system. This way, we can get the estimate

k∂ t U _err ^ε k _L

2

(Ω)

³

+ k∇∂ t U _cor ^ε k _L

2

(Ω

^εA

)

^3×3

+ εk∇∂ t U _cor ^ε k _L

2

(Ω

^εA

)

^3×3

≤ C √

ε + ε + kΘ _err k _L

2

(Ω) + k∂ t Θ _err k _L

2

(Ω)
. (4.11) Since this is done quite analogously to the estimates leading to inequality (4.10), except for a few terms arising due to the time differentiation, the details of this are omitted.

If we take ∂ _t U _cor

ε

as a test function and follow the same strategy as in (4.4), we can estimate

Z t 0

I ₁ ^ε (τ, ∂ _t U _cor0 ^ε ) dτ − Z t

0

Z

Ω

^ε_A

C _A ^r e _y (τ ^u ) 

ε θ _A : e(∂ _t U _cor0 ^ε ) dx dτ

≥ C ₁ 

ke(U _cor ^ε )(t)k ² _L

2

(Ω

^ε_A

) − ke(U _cor ^ε )(0)k ² _L

2

(Ω

^ε_A

)



− C ₂ Z t

0

kU _cor ^ε k ² _H

1

(Ω

^εA

)

³

dτ − C ₃ (ε + ε ² ),

(4.12)

where integration by parts w.r.t. time was done. Similarly, we obtain Z t

0

I ₂ ^ε (τ, ∂ t U _cor0 ^ε ) dτ

≥ C ₁ ε ² 

ke(U _cor ^ε )(t)k ² _L

2

(Ω

^εB

)

^3×3

− ke(U _cor ^ε )(0)k ² _L

2

(Ω

^εB

)

^3×3



− C ₂ ε ² Z t

0 ke(U _cor ^ε )k ² _L

2

(Ω

^εB

)

^3×3

dτ − C ₃ ε ² .

(4.13)

Moreover,

9

X

i =5

|I _i ^ε (t, ∂ _t U _cor0 ^ε )| ≤ Cε 

kU _cor ^ε k _H

1

(Ω

^ε_A

)

³

+ kU _cor ^ε k _L

2

(Ω

^ε_B

)

³



. (4.14)

4.2. Estimates for the heat conduction equations. Now we go on with estab- lishing some control on the error terms in the heat conduction equations. Multi- plying equations (4.1a) and (4.1c) with test functions ϕ ∈ H ₀ ¹ (Ω), integrating over Ω, and then integrating by parts while using the interface conditions, we are lead to

Z

Ω

^εA

∂ t



c ^r,ε _A Θ ^ε _A − κ A

Z

Y

A

c ^r dy θ A

 ϕ dx

| {z }

=:E

1^ε

(t,ϕ)

+ Z

Ω

^εA

K _A ^r,ε ∇Θ ^ε _A − κ A K _A ^h ∇θ A
· ∇ϕ dx

| {z }

=:E

2^ε

(t,ϕ)

+ Z

Ω

^ε_A

((c ^r,ε _A Θ ^ε _A + γ _A ^r,ε : ∇U _A ^ε ) v ^r,ε ) · ∇ϕ dx

| {z }

=:E

^ε3

(t,ϕ)

+ Z

Ω

^εA

K _A ^r,ε ∇Θ ^ε _A − κ A K _A ^h ∇θ A
· ∇ϕ dx − Z

Γ

^ε

κ A K _A ^h ∇θ A · n ^ε ϕ dx

| {z }

=:E

^ε4

(t,ϕ)

+ Z

Ω

^εB

∂ t (c ^r,ε _B Θ ^ε _B ) − ∂ t (c ^r _B Θ B ) 

ε
ϕ dx

| {z }

=:E

^ε6

(t,ϕ)

+ Z

Ω

^ε_B



∂ _t (εγ _B ^r,ε : ∇U _B ^ε ) − h

∂ _t (γ _B ^r : ∇ _y U _B ) i

ε

 ϕ dx

| {z }

=:E

^ε6

(t,ϕ)

+ Z

Ω

^εB

c ^r,ε _B Θ ^ε _B v ^r,ε − c ^r _B Θ B



ε v ^r 

ε
· ∇ϕ dx

| {z }

=:E

7^ε

(t,ϕ)

+ ε Z

Ω

^εB

γ _B ^r,ε : ∇U _B ^ε v ^r,ε − γ _B ^r 

ε : ∇[U B ] ε [v ^r ] ε
· ∇ϕ dx

| {z }

=:E

8^ε

(t,ϕ)

+ ε ² Z

Ω

^ε_B

K _B ^r,ε ∇Θ ^ε _B − K _B ^r 

ε ∇Θ B



ε
· ∇ϕ dx

| {z }

=:E

^ε9

(t,ϕ)

= − Z

Γ

^ε

W _Γ ^r,ε ϕ dσ + Z

Ω

^εA

κ A W _Γ ^h ϕ dx

| {z }

=:E

10^ε

(t,ϕ)

+ Z

Ω

^εA

κ A

Z

Γ

K _B ^r ∇ y Θ B · n ds ϕ dx − ε ² Z

Γ

^ε

[K _B ^r ] ε ∇[Θ B ] ε · n ^ε ϕ dσ

| {z }

=:E

11^ε

(t,ϕ)

+ Z

Ω

^ε_A

 f _θ ^r,ε

A

− κ A

Z

Y

_A

f _θ ^r

A

dy

 ϕ dx +

Z

Ω

^ε_B

f _θ ^r,ε

B

− f _θ ^r

_B



ε
ϕ dx

| {z }

=:E

12^ε

(t,ϕ)

+E ^ε ₁₃ (t, ϕ)