AND TRAINING OF SCIENCE AND TECHNOLOGYGRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY Nguyen Quang Huy A SPACE – TIME FINITE ELEMENT METHOD FOR AN ADVECTION – DIFFUSION PROBLEM WITH A MOVI
Functional spaces
Sobolev spaces
In this section, we considerΩ to be an open set inR d (d = 1,2,3).
Definition 1.6 (Lebesgue space) Let q ∈ [1,∞) The spaceL q (Ω)consists of all Lebesgue measurable functions u, defined at almost everywhere in Ω such that
The spaceL ∞ (Ω) includes all Lebesgue measurable functions u, defined at almost everywhere inΩ that satisfing
Definition 1.7 (Integer order Sobolev space) Let s ∈ N, q ∈ [1,∞] We define the spaceW s,q (Ω)as
W s,q (Ω) := {u∈ L q (Ω)| ∂ α u ∈ L q (Ω) for all0 ≤ |α| ≤ s}, where∂ α udenotes theα-order weak derivative ofu, endowed with the norm
Definition 1.8 (Fractional order Sobolev space) Let s > 0, s /∈ N and q ∈
[1,∞) The space W s,q (Ω)consists of all functions u ∈ W ⌊s⌋,q (Ω)such that
|x−y| d+q(s−⌊s⌋) dxdy < ∞, equipped with the norm
Theorem 1.9 Let s ≥ 0 The space W s,q (Ω) furnished with the norm
∥ã∥ W s,q (Ω) is a Banach space for all q ∈ [1,∞] In particular, the space
W s,2 (Ω) is a Hilbert space, denoted by H s (Ω) The norm on this space is denoted by∥ã∥ H s (Ω)
Theorem 1.10 (Embedding and compact embedding) Let s > 0, q ∈ [1,∞] andΩ be a Lipschitz domain inR d (d = 1,2,3). a) If sq > d then we have the embedding W s,q (Ω) ,→ L ∞ (Ω) and the compact embedding W s,q (Ω) ,→ C Ω
. b) If sq ≤ d then for all q ′ ∈ h
, we have the compact embedding
Anisotropic Sobolev spaces
In this section, we consider Ω to be an open set in R d (d = 1,2,3) and
T > 0 to be a positive number Denote by Q T := Ω×(0, T) a space-time domain.
Definition 1.11 (t-anisotropic Sobolev space) Let l, k ∈ N We define the spaceH l,k (Q T ) as
H l,k (Q T ) :u ∈ L 2 (Q T ) | ∂ x α ∂ t r u ∈ L 2 (Q T ) for all 0≤ |α| ≤ l, r = 0,1, , k , where ∂ x α u and ∂ t r u are the weak derivatives with respect to x and t of u, respectively, endowed with the norm
LetH 1,0 0 (Q T )be the closure of the spaceC 1 0 (Q T )with respect to the norm
T ) For convenience, we use the compact notation Y := H 1,0 0 (Q T ), furnished with an equivalent norm
Ω κ|∇u| 2 dxdt, whereκ = κ(x, t) will be introduced in chapter 2 The equivalence results from the Poincar´e–Steklov inequality
T ) ∀u ∈ Y, (1.1) we refer to [20] The dual space of Y is denoted by Y ′ (see Definition1.18), and the duality pairing betweenY ′ andYis denoted by⟨ã,ã⟩ Let us introduce the spaces
X t := {u ∈ X| u(ã, t) = 0}, witht ∈ {0, T}, equipped with the norm∥u∥ 2 X := ∥u∥ 2 Y +∥∂ t u∥ 2 Y ′
Theorem 1.12 (Time trace and integration by parts) Let X be the space defined in(1.2) Then, the following statements hold true a) The spaceX is embedded into the spaceC [0, T],L 2 (Ω)
. b) The trace operator u ∈ X → u(ã, t) ∈ L 2 (Ω) is bounded for almost every t ∈ [0, T] The following inequality holds sup t∈[0,T ]
Consider the case Q T is separated into two subdomains Q 1 and Q 2 by the space-time interface Γ ∗ := ∂Q 1 ∩ ∂Q 2 In this scenario, the space
H 1,0 (Q1 ∪Q2) is important, since in it, the trace operators γi : H 1,0 (Qi) →
The Stein extension operators are essential for managing functions that exhibit global low regularity but local high regularity Under mild assumptions on Γ ∗, the operators L 2 (Γ ∗) for i = 1,2 are well-defined For any fixed s ≥ 0, a function u belonging to H s (Q 1 ∪ Q 2) can be expressed as u i := u | Q i for i = 1,2.
H s (Q i ) the restriction of uto the subdomain Q i
Theorem 1.13 Assume that Γ ∗ is a Lipschitz continuous hypersurface in
R d+1 , then there exists smooth extensionsE i : H s (Q i ) →H s (Q T )such that
Bounded linear operators
Weak convergence in Banach spaces
Definition 1.20 (Weak convergence) LetUbe a normed space We say that the sequence {u n } n∈
N ⊂ U converges weakly to u ∈ U, written as u n ⇀ u inU, if S(u n ) →S(u) for allS ∈ U ′
Theorem 1.21 Let U be a normed space Then, every closed and convex subset E of U is weakly sequentially closed, which means for any sequence {u n } n∈
N ⊂ E such thatu n ⇀ uinU, we implyu ∈ E.
Theorem 1.22 (Weak convergence in a reflexive space) Let Ube a reflexive Banach space Then, every bounded, closed and convex subset E of U is weakly sequentially compact, that is, for any sequence {u n } n∈
N such that there exists u ∈ E satisfying un ⇀ u inU.
Theorem 1.23 (Weak convergence with convex continuous functional) LetU be a Banach space Then, every convex continuous functional S : D(S) ⊂
U → R is weakly lower semicontinuous, which means for any sequence
N ⊂ U such that there exists u ∈ U satisfying un ⇀ u in U, we havelim inf n→∞ S(u n ) ≥ S(u).
Differentiability in Banach spaces
ConsiderU,W as two Banach spaces andE as an open subset of U.
Definition 1.24 (Gˆateaux derivative) An operator S : E → W is said to have a directional derivative in the direction h ∈ U at an element u ∈ E, written asDS(u, h), if there exists the limit τ→0lim
If the operatorh → DS(u, h) is a bounded linear operator, then we say that
S is Gˆateaux differentiable at u ∈ E with the Gˆateaux differential DS(u, h) and the Gˆateaux derivativeS ′ given byDS(u, h) =: S ′ (u) (h).
Definition 1.25 (Frech´et derivative) A continuous operator S : E → W is said to be Frech´et differentiable at an elementu ∈ Eif there esists a bounded linear operatorDS(u) : U →W such that h→0lim
The termDS(u) (h)is refered as the Frech´et differential of S atu ∈ E with the variation h ∈ U, and DS(u) is the Frech´et derivative of S at u ∈ E, denoted byS ′ (u).
Clearly, if an operator is Frech´et differentiable operator, then it is Gˆateaux differentiable Moreover, in that case, these two derivatives coincide.
An advection-diffusion equation with a moving interface
This chapter presents the interface-fitted space-time finite element method
The advection-diffusion equation with a moving interface is crucial for addressing complex problems in engineering and physical phenomena involving multi-component systems This equation is relevant in various applications, including mass transport, heat transfer, electromagnetics, and heat induction.
U in (2.1) may represent the concentration of the pollutant or the electron transported at a velocityvowing to the advection and diffusion effect.
Let Ω be a Lipschitz domain in R^d (where d = 1 or 2) with boundary ∂Ω This domain is divided into two time-dependent subdomains, Ω₁(t) and Ω₂(t), separated by an interface Γ(t) for all t in the interval [0, T], where T is greater than 0 The interface Γ(t) is moved by a velocity field v = v(x, t), which is continuous over [0, T] and twice continuously differentiable in Ω.
∇ãv(x, t) = 0for all(x, t) ∈ Ωì[0, T][25] We denote byQ T := Ωì(0, T) the space-time domain and
Ωi(t)× {t} (i = 1,2) two subdomains seperated by the space-time interfaceΓ ∗ := S t∈(0,T )Γ(t)× {t} Assume thatΓ ∗ is aC 2 -regular hypersurface inR d+1 andΓ(t)∩∂Ω = ∅ for allt∈ [0, T] Consider the following problem
(2.1) whereF is the source term, U 0 is the initial value, and n stands for the unit normal atΓ(t) pointing from Ω1(t) into Ω2(t) The notation [U] = U 1 | Γ(t) −
U 2 | Γ(t) denotes the jump ofU acrossΓ(t), withU i | Γ(t) the limiting value from
Figure 2.1: The interface Γ(t), which envolves by a velocity v, devides the domain Ω into two subdomains Ω 1 (t) and Ω 2 (t), consider the case d = 1 [5].
For simplicity, let us assume that the diffusion coefficient κ is a positive constant on each subdomain κ
The general setting of the subdomain-wise continuous uniformly positive co- efficient κ ∈ L ∞ (QT) can be treated similarly In this chapter, the constant
C > 0depends on the space-time domainQ T , the position of the space-time interfaceΓ ∗ , the norm∥v∥ L ∞ (Q
T ), and the coefficientκ, but is independent of the functionu, the function u, and the mesh size h Their different values in different contexts are allowed.
Variational formulation
In this section, we revisit the variational formulation of problem (2.1) and its well-posedness, as outlined in [5] We consider F belonging to Y' and U0 in H1₀(Ω), with u0 being an extension of U0 in X The solution to problem (2.1) is defined as U = u + u0 in X, where u in X0 satisfies the equation a(u, φ) = ⟨F, φ⟩ - a(u0, φ) for all φ in Y, with the bilinear form a: X × Y → R defined as a(u, φ) := ⟨∂t u, φ⟩ +
Lemma 2.1 There exists a constantC > 0that satisfies sup φ∈Y \{0} a(u, φ)
∥φ∥ Y ≥C ∥u∥ X ∀u ∈ X 0 Lemma 2.2 If a(u, φ) = 0for all u ∈ X 0 thenφ = 0.
The well-posedness of the problem (2.2) results from Lemmas2.1and2.2, according to the Banach-Neˇcas-Babuˇska theorem [26].
Theorem 2.3 LetF ∈ Y ′ and U 0 ∈ H 1 0 (Ω) Then, the problem(2.2) admits a unique solutionu ∈ X 0 such that∥u∥ X ≤ C(∥F∥ Y ′ + ∥u 0 ∥ X ).
Remark 2.4 If F ∈ L 2 (Q T ) then we have a priori estimate
, (2.3) since from the inequality (1.1), it holds that
Regarding an additional regularity of the solution U ∈ X of the problem (2.1), let us introduce the following assumption:
Assumption 2.1 For F ∈ Y ′ and U 0 ∈ H 1 0 (Ω), assume that the solution
U ∈ X of the problem (2.1) satisfies U ∈ H 1 (Q T ) ∩ H s (Q 1 ∪Q 2 ) with a given s > d+3 2 and there exists a constant C > 0 independent of F and U0 such that
In this work, we assume that the assumption2.1 is satisfied.
Finite element discretization
Interface-fitted space-time method
We employ the interface-fitted space-time finite element method to discretize the problem outlined in (2.2), following the framework established in [5] For ease of analysis, we set the initial condition U 0 to zero, which implies that u 0 equals zero in (2.2) Additionally, we define Y h as the finite element space consisting of continuous, element-wise linear functions on T h, constrained to have zero values at specific points.
Y h ⊂ Y and X h,0 ⊂ X 0 Consider the discrete problem: Find u h ∈ X h,0 that satisfies a h (u h , φ h ) =⟨F, φ h ⟩ ∀φ h ∈ Y h , (2.5) with the bilinear forma h : X 0 ×Y →Rgiven by ah(u, φ) = ⟨∂ t u, φ⟩+
(vã ∇u)φ+κh∇uã ∇φdxdt, whereκ h approximates κby means of κ h :
κ 1 > 0 in Q 1,h , κ 2 > 0 in Q 2,h Regarding the numerical analysis, we introduce the seminorm
Note that in casev ∈ Y ⊂ H 1,0 (Q 1,h ∪Q 2,h ), the right-hand side becomes
The equivalent norm in the space Y is defined as Ω κ h |∇v| 2 dxdt, represented as |||v||| This norm, which incorporates the coefficient κ h, is more advantageous for analyzing discrete problems compared to the standard norm ∥v∥ Y Additionally, we define a new norm on the space H 1 (Q 1,h ∪ Q 2,h ).
|||v||| 2 ∗ := |||v||| 2 +|||z h (v)||| 2 ∀v ∈ H 1 (Q 1,h ∪Q 2,h ), wherez h (v) ∈ Y h is a unique solution of the problem
Lemma 2.5 There exists a constantC > 0such that sup φ h ∈Y h \{0} a h (u h , φ h )
|||φ h ||| ≥ C|||u h ||| ∗ ∀u h ∈ X h,0 (2.6)Using the discrete Banach-Neˇcas-Babuˇska theorem [26], we conclude that the problem (2.5) is uniquely solvable.
Auxiliary results
In this section, we provide some auxiliary findings We first present a result regarding the mismatch between each space-time subdomain Q i and its approximated counterpartQ i,h , fori = 1,2 Define by
S h 1 := Q 1,h \Q 1 = Q 2 \Q 2,h , S h 2 := Q 2,h \Q 2 = Q 1 \Q 1,h ,andS h = S h 1 ∪S h 2 (see Figure2.2) We denoteT h ∗ = {K ∈ T h | K ∩Γ ∗ ̸= ∅} the set of all interface elements.
Figure 2.2: The mismatch region S h = S h 1 ∪ S h 2 lies between the space-time interface Γ ∗ and the discrete one Γ ∗ h , consider the case d = 1 [5].
Lemma 2.6 Assume that Γ ∗ is a C 2 -continuous hypersurface in R d+1 (d 1,2) andT h is a quasi-uniform mesh Then, for eachK ∈ T h ∗ , we have
It holds for the cardinality of the setT h ∗ that
Proof When d = 1, the proof of the first inequality can be found in [27].
We obtain the second one by combining this inequality with [28] All the arguments can be extended to the cased = 2without essential changes.
We continue by studying the approximability of the Lagrangian inter- polant Let u ∈ H 1 (Q T ) ∩ H 2 (Q 1 ∪Q 2 ) For u ∈ H 2 (Q 1 ∪Q 2 ), the Sobolev embedding [20] follows that u ∈ C Q1
In addition, if u ∈ H 1 (Q T ), then γ 1 u − γ 2 u = 0, which implies u ∈ C Q T
The nodal interpolation operator, denoted as X h,0, has been explored in the context of interpolation estimates by Chen and Zou when d = 1 Their findings indicated that the order of the estimate was nearly optimal, differing only by a factor of |logh|, with h representing the mesh size This paper introduces an additional condition on the function u and builds upon their methodology to achieve an optimal order estimate, leading to significant results.
Lemma 2.7 Foru ∈ H 1 (Q T )∩H s (Q 1 ∪Q 2 )withs > d+3 2 , the interpolation operatorI h satisfies the following inequality
1 ∪Q 2 ), (2.9) whereD := (∇, ∂ t ) ⊤ denotes the space-time gradient operator.
Proof Let us focus on ∥u−I h u∥ L 2 (Q
To estimate the interpolation error for each element \( K \in T_h \), we first evaluate the error individually for each element and then aggregate these errors across all elements Assuming \( u \in H^2(K) \) for any \( K \notin T_h^* \), classical interpolation theory provides the necessary framework for this estimation.
Consider an arbitrary element K in T h ∗, where K intersecting S h is a subset of Q 1 and K excluding S h is a subset of Q 2 For a function u in H s (Q 1 ∪ Q 2) with s greater than (d + 3)/2, it is important to note that E i u belongs to H s (Q T) and is also included in W 1,∞ (Q T) for i = 1, 2, as indicated by extension operators E 1 and E 2 The inequality (2.7) combined with classical interpolation theories leads to significant conclusions.
We sum over allK ∈ T h ∗ and use the inequality (2.8), the Sobolev embedding
H s−1 (QT) ,→ L ∞ (QT) for s > d+3 2 [20], and the extension operators (1.4) again to get
1 ∪Q 2 ) Using the same arguments, one obtains ∥D (u−I h u)∥ L 2 (Q
Please note that the mismatch betweenT h and Q T at the interface leads to the non-conformal property of ah(ã,ã) In particular, we have the following lemma:
Lemma 2.8 Letu ∈ X 0 andu h ∈ X h,0 be the solutions of the problems(2.2) and (2.5), respectively There holds the following equality a h (uưu h , φ h ) Z
(κ h −κ)∇uã ∇φ h dxdt ∀φ h ∈ Y h (2.12)Proof For u ∈ X 0 , u h ∈ X h,0 and φ h ∈ Y h , we invoke the equations (2.2) and (2.5) to have ah(u, φh) =⟨∂ t u, φh⟩+
(κ h −κ)∇uã ∇φ h dxdt, observed thatκh−κvanishes everywhere outside of Sh.
A priori error estimates
In this section, we analyze the error \( u - u_h \) across different norms, where \( u \in X_0 \) and \( u_h \in X_{h,0} \) represent the solutions to problems (2.2) and (2.5) Our findings focus on the error \( u - u_h \) within the norm \( ||| \cdot |||^* \), as detailed in reference [5].
Lemma 2.9 Letu ∈ X 0 andu h ∈ X h,0 be the solutions of the problems(2.2) and (2.5), respectively Assume that Assumption 2.1 is satisfied Then, we have the following estimate
We continue by estimating the state error in the L 2 (Ω)-norm at t = T. Following the duality argument, let us define the space
V T :n ψ ∈ X | ψ(ã, T) =γ T ∥(uưu h ) (ã, T)∥ ư1 L 2 (Ω)(uưu h ) (ã, T) inΩ o, where γ T > 0 is a sufficient large number, u ∈ X 0 and u h ∈ X h,0 are the solutions of the problems (2.2) and (2.5), respectively Assume that there existsy ∈ V T that solves the problem
Under the assumption2.1, one hasy ∈ H 1 (Q T )∩H s (Q 1 ∪ Q 2 )withs > d+3 2
We further assume that it satisfies
1 ∪Q 2 ) ≤ C, (2.15) where the constantC > 0is independent of uand uh.
Theorem 2.10 Let u ∈ X 0 and u h ∈ X h,0 be the solutions of the problems (2.2)and (2.5), respectively Assume that Assumption2.1 is satisfied and the problem(2.14) admits a solution y ∈ V T that satisfies(2.15) Then, we have
Proof We chooseϕ = uưu h ∈ X 0 in (2.14), then employ the integration by parts formula to get γ T ∥(uưu h ) (ã, T)∥ L 2 (Ω)
In the analysis of the equation Ω ư(vã ∇y) (uưuh) + κ∇(uưuh)ã ∇ydxdt, we utilize the condition (uưu h)(ã,0) = 0 within the domain Ω To address the advection component on the right-hand side, we apply the property ∇ãv(x, t) = 0 for all (x, t) in the domain Ω × [0, T], along with the divergence theorem and the concept of homogeneity.
Dirichlet boundary condition We have
(vã ∇(uưu h ))y +y(uưu h ) (∇ ãv) dxdt
(vã ∇(uưu h ))ydxdt, (2.17) withn Ω the outward normal to∂Ω Hence, we get γ T ∥(uưu h ) (ã, T)∥ L 2 (Ω) = ⟨∂ t (uưu h ), y⟩
(κưκ h )∇(uưu h )ã ∇ydxdt, noticing thatκ−κ h = 0outside ofS h Sincey ∈ H 1 (Q T )∩H s (Q 1 ∪Q 2 )for s > d+3 2 , we are able to invoke the interpolationI h y We denotee := y−I h y for convenience, and chooseφh = Ihy ∈ Xh,0 in (2.12) to obtain γ T ∥(uưu h ) (ã, T)∥ L 2 (Ω) = a h (uưu h , y) +
(κưκ h ) (∇uã ∇eư ∇uã ∇y +∇(uưu h )ã ∇y) dxdt (2.18)
We first estimate the discrete bilinear term In doing so, we integrate by parts again, note that u − u h ∈ X 0 and e ∈ H 1 (Q T ), and apply the inequalities (2.9), (1.1), (2.13), and (2.15) One has ah(uưuh, e)
By using the trace inequality (1.3), the inequality (1.1), the estimate (2.9), and the inequality (2.15), we observe that
1 ∪Q 2 ) ≤ Ch, (2.19) which yields ah(uưuh, e) ≤ Ch∥(uưuh) (ã, T)∥ L 2 (Ω) + Ch 2 ∥u∥ H s (Q
Next, consider the second integral on the right-hand side of (2.18), denoted byI for short By using the Cauchy-Schwarz inequality, together with (2.9) and (2.13), we boundI by
(κưκ h ) (∇uã ∇eư ∇uã ∇y+ ∇(uưu h )ã ∇y) dxdt
We follow the arguments of (2.11) to estimate ∥∇u∥ L 2 (S h ) and ∥∇y∥ L 2 (S h ).
Take∥∇u∥ L 2 (S h )for instance Under Assumption2.1, we haveu ∈ H 1 (Q T )∩
1 ∪Q 2 ). Similarly, for y ∈ H 1 (QT) ∩ H s (Q1 ∪Q2) with s > d+3 2 , we can prove that ∥∇y∥ L 2 (S h ) ≤ Ch∥y∥ H s (Q
1 ∪Q 2 ) ≤ Ch by using the inequality (2.15).
We substitute (2.20) and (2.21) into (2.18) to arrive at
1 ∪Q 2 ). Since h ∈ (0, h ∗ ) for a given h ∗ > 0, the proof is finished by choosing γ T such thatγ T ≥ Ch ∗ + 1.
Using the inequality (2.16), we are now able to estimate ∥uưu h ∥ L 2 (Q
T ), whereu ∈ X0 anduh ∈ Xh,0 be the solutions of the problems (2.2) and (2.5), respectively In the following lemma, assume that there exists a solution z ∈ Xto the problem
(uưu h )ϕdxdt ∀ϕ ∈ Y, (2.22) withz(ã, T) ∈ H 1 0 (Ω) The assumption2.1yieldsz ∈ H 1 (QT)∩H s (Q1 ∪Q2) withs > d+3 2 Moreover, assume that
1 ∪Q 2 ) ≤C, (2.23) for a constantC > 0independent of uand u h
Theorem 2.11 Let u ∈ X0 and uh ∈ Xh,0 be the solutions of the problems (2.2) and (2.5), respectively Assume that the assumption of lemma 2.10 is satisfied and there exists a solutionz ∈ Xof the problem(2.22) that satisfies (2.23) Then, there holds the following estimate
Proof We choose ϕ = uưu h ∈ X 0 in (2.22), then apply the integration by parts formula, the arguments of (2.17), and the equality (2.12) to arrive at
In the equation (uưu h)(x, T)z(x, T) dx, we apply the initial condition (uưu h)(ã,0) = 0 in Ω, defining e ′ as z − I h z The interpolation I h z can be utilized since z belongs to the space H 1(Q T) ∩ H s(Q 1 ∪ Q 2) with s > (d + 3)/2 To analyze the term a h(uưu h, e ′), we integrate by parts, considering uưu h ∈ X 0 and e ′ ∈ H 1(Q T), while leveraging the inequalities (2.9), (1.1), (2.13), and (2.23) to derive a h(uưu h, e ′).
On the other hand, by following the technique as in (2.21), one can show that
(κ h ưκ) (∇uã ∇zư ∇uã ∇e ′ ư ∇(uưu h )ã ∇z) dxdt
By substituting (2.26) and (2.27) into (2.25), we imply
1 ∪Q 2 )∥(I h z) (ã, T)∥ L 2 (Ω) , invoking the estimate (2.16) in the final line.
The last step is to estimate ∥(I h z) (ã, T)∥ L 2 (Ω) We apply the technique as in (2.19), the inequality (1.1) with I h z ∈ X h,0 and theH 1 -seminorm stability of the interpolation operatorI h [20] One gets
T ). The proof is finished by employing a priori estimate (2.4) for the problem (2.22) (after changing the time and the velocity field directions) We have
An inverse source problem for the advection-diffusion equation with a moving interface
This chapter focuses on investigating an inverse source problem related to the advection-diffusion equation characterized by a moving interface We consider the scenario where the initial value \( U_0 \) is zero, and the source term \( F \) is defined in a specific form.
F (x, t) = ℓ(x, t)f (x, t) +g(x, t) for all (x, t) ∈ Q T , whereℓ ∈ L ∞ (Q T ) and g ∈ L 2 (Q T ) are given Moreover, assume that there exists a constant
L > 0 such that ℓ ≥ L at almost everywhere in QT Let U be the solu- tion of this problem Determine f ∈ L 2 (Q T ), given a partial interior data
The equation U_d := U | ω T, along with the condition f ≥ 0 almost everywhere in QT, establishes the framework for our analysis Given that U0 = 0, it follows that u0 = 0, leading to the conclusion that the solution u ∈ X0 can be expressed as the sum of two components: u = u + u∗ Here, u ∈ X0 represents the solution to the variational problem defined by a(u, φ) = (ℓf, φ) in L²(Q).
T ) ∀φ ∈ Y, (3.1) andu ∗ ∈ X 0 solves the variational problem a(u ∗ , φ) = (g, φ) L 2 (Q
The functionu ∗ is uniquely determined We aim to identifyf ∈ F+ in (3.1) from the partial interior data U d ∈ L 2 (ω T ) in the subdomain ω T , where the admissible set is defined by
This set is non-empty, close and convex Our inverse source problem reads as the following operator equation with a priori information
T ∈ L 2 (ω T ) is the exact data and A is the bounded linear operator, defined by
T Here, we use the notation u(f) to emphasize the dependence of u in (3.1) on f For avoiding ambiguity, we interpret z d ≡ 0 in Q T \ ω T so that it is well-defined inL 2 (QT).
In this chapter, C > 0is a constant dependent on the space-time domain
Q T , the position of the space-time interface Γ ∗ , the norm ∥v∥ L ∞ (Q
The function ℓ, the function g, and the coefficient κ are influenced by T but remain independent of the parameter λ, the noise level ε, the regularized state u ε λ, the adjoint p ε λ, the sources f + and f λ ε, and the mesh size h Variations in their values are permissible across different contexts.
The ill-posedness of the problem
Let us discuss the concept of solutions to the problem (3.4) Firstly, this problem may not have solutions since z d can be outside the restricted range
The operator A is generally not injective, as demonstrated by examples where two solutions of problem (3.1) coincide in subdomain Q1 but differ in subdomain Q2 Consequently, problem (3.4) may have multiple solutions, highlighting the importance of understanding this concept as defined in [29].
In the context of the admissible set F + defined in (3.3), an element f + is identified as the F+-best approximated solution to the problem outlined in (3.4) if it possesses the lowest L 2 (Q T )-norm among all elements f within F + that satisfy the problem's requirements.
The continuous unregularized source, denoted as f + ∈ F +, is uniquely determined Although A is a linear operator, the presence of an inequality constraint renders the problem nonlinear, making the ill-posedness criterion for linear problems inapplicable Consequently, we reference the local ill-posedness concepts for nonlinear problems as outlined in [31].
Definition 3.2 Letf ∈ F+ be a solution of the problem (3.4) The problem (3.4) is said to be locally well-posed at f ∈ F + if there exists a closed ball
B r (f) ⊂ L 2 (Q T ) with the center f ∈ F + and radius r > 0 such that for every sequence{f n } n∈
N ⊂ F+∩ Br(f), if lim n→∞∥Af n −Af∥ L 2 (ω
T ) = 0 Otherwise, the problem (3.4) is said to be locally ill-posed atf ∈ F +
The compact embedding H 1,0 (QT) ,→ L 2 (QT) [20] implies A is a com- pact operator Together with the arguments in [32], we conclude that the problem (3.4) is locally ill-posed at every point inF +
Tikhonov regularization
The ill-posed nature of the problem (3.4) indicates that its approximate solution lacks continuous dependence on the data, necessitating regularization to achieve a stable solution In this study, we utilize Tikhonov regularization to reformulate the problem as min f ∈F +.
In this study, we consider T ∈ L 2 (ω T) as the noise data, with λ > 0 representing the regularization parameter Following the approach outlined in problem (3.4), we interpret z d ε as an element of L 2 (Q T), defining z d ε ≡ 0 outside of ω T Additionally, for a given noise level ε > 0, we define U d ε ∈ L 2 (ω T) as the imprecise observation of the data.
Theorem 3.3 The regularized problem(3.5) has a unique solutionf λ ε ∈ F+. Proof Clearly, the set F + := {f ∈ F+ | the problem (3.1) is well-posed} is non-empty Together withJ λ ε (f) ≥0onF + , we deduce thatj := inf f ∈F +
J λ ε (f) is finite Hence, there exists a sequence{f n } n∈
T ) ≤ 2 λ J λ ε (f n )for all n∈ Nimplies that the sequence {f n } n∈
N is bounded inL 2 (Q T ), which allows us to extract a (not relabeled) weakly convergent subsequence{f n } n∈
N such that fn ⇀ f λ ε inL 2 (QT) with f λ ε ∈ L 2 (Q T ) Moreover, there exists a sufficiently larger > 0such that
N ⊂ F + ∩Br, where B r denotes a closed ball with the radius r > 0 in L 2 (Q T ) Since
F + ∩ B r is a closed, bounded, and convex subset of L 2 (Q T ), it is weakly sequentially compact [22] This gives usf λ ε ∈ F +
Consider the variational problem: Find u n := u(f n ) ∈ X 0 that satisfies a(u n , φ) = (ℓf n , φ) L 2 (Q
This problem is well-posed A priori estimate (2.3) says
N is bounded in X Hence, there exists u ε λ ∈ Xand a (not relabeled) weakly convergent subsequence {u n } n∈
N such thatu n ⇀ u ε λ inX Therefore, for anyφ ∈ Y, we have n→∞lim
By passing the limit into (3.6), we arrive at a(u ε λ , φ) = (ℓf λ ε , φ) L 2 (Q
To conclude that u ε λ = u(f λ ε ), one needs to prove u ε λ ∈ X 0 In (3.6), we chooseφ ∈ C [0, T],H 1 0 (Ω) with φ(ã, T) = 0 in Ω and integrate by parts to get
T ), (3.8) sinceu n (ã,0) = 0inΩ We take n → ∞to obtain
On the other hand, by applying the technique in (3.8), we rewrite (3.7) as follows
From the last two equations, we imply u ε λ (ã,0) = 0 in Ω, and hence u ε λ u(f λ ε ) Therefore, we get j = lim inf n→∞ J λ ε (f n ) = lim inf n→∞
= J λ ε (f λ ε ), which indicates that f λ ε ∈ F + is a minimizer The uniqueness follows from the strict convexity of the functionalJ λ ε The proof is complete.
Next, we derive the optimality conditions of the regularized problem (3.5).
In doing so, let us introduce the following adjoint problem: Identifyp(f) ∈
T ) ∀ϕ ∈ Y, (3.9) where the bilinear forma ′ : X×Y → Ris defined by a ′ (p, ϕ) = − ⟨∂ t p, ϕ⟩+
−(vã ∇p)ϕ+κ∇pã ∇ϕdxdt, andχ ω T is the characteristic function of the subdomain ω T By changing the time and the velocity field directions, and applying [5] withχ ω T (u(f)−z d ε ) ∈
L 2 (Q T ), we conclude the well-posedness of this problem.
Theorem 3.4 The unique solution f λ ε ∈ F + of the problem (3.5), together with the corresponding state u ε λ ∈ X 0 and adjoint p ε λ ∈ X T , satisfies the following optimality conditions a(u ε λ , φ) = (ℓf λ ε , φ) L 2 (Q
Proof Following the classical arguments [22], we show that the functionalJ λ ε defined by (3.5) is Fr´echet differentiable and its gradient∇J λ ε (f) atf ∈ F + has the form
∇J λ ε (f) =ℓp(f) +λf, with p(f) ∈ X T solves the problem (3.9) Indeed, take a small variation δf ∈ L 2 (QT) off ∈ F+, we have
T ). Here, we know thatu(δf) ∈ X0 is the solution of the problem a(u(δf), φ) = (ℓδf, φ) L 2 (Q
Owing to the inequality (1.1) and a priori estimate (2.4), one has
To derive the functional gradient, we express the first term on the right-hand side of equation (3.14) as a scalar product within the solution space By letting p(f) belong to X T as the solution of equation (3.9), we select ϕ = u(δf) from X 0 in (3.9) to proceed with our analysis.
By integrating by parts, we can rewrite⟨∂ t p(f), u(δf)⟩ as
= − ⟨∂ t u(δf), p(f)⟩, usingp(f) ∈ X T and u(δf) ∈ X 0 in the final step To handle the advection part on the right-hand side of (3.15), we invoke the technique as in (2.17).
Here, we choose φ = p(f) ∈ X T in (3.13) to obtain the first equality By substituting into (3.14), we can conclude the Fr´echet differentiability of the functionalJ λ ε , together with its gradient.
This section concludes by discussing the convergence properties of Tikhonov regularization, highlighting the stability of the solution \( f_{\lambda, \epsilon} \in F^+ \) concerning noise in the observation \( z_{d, \epsilon} \in L^2(Q_T) \) The theorem presented here serves as a constrained variant of the findings from previous studies [33] and [34].
N ⊂ L 2 (ω T ) be the sequence that converges strongly toz d ε in L 2 (ω T ), and {f n } n∈
N ⊂ F + the sequence of solutions to the corresponding problems fmin∈F +
N converges strongly to the solution f λ ε ∈ F+ of the problem (3.5)inL 2 (Q T ).
Proof Owing to Theorem 3.3, for each n ∈ N, there exists a unique mini- mizerf n ∈ F + of the problem (3.17) For allf ∈ F + , we have
T ), which implies the boundedness of {f n } n∈
N in L 2 (Q T ) By following the technique as in Theorem 3.3, we conclude the existence of an element f λ ε ∈
F + and a (not relabed) weakly convergent subsequence {f n } n∈
N such that f n ⇀ f λ ε in L 2 (Q T ) Moreover, as n→ ∞, it holds u(f n ) ⇀ u(f λ ε ) in X, up to taking a further subsequence Together with the strong convergence of the sequence{z n } n∈
N toz d ε inL 2 (ω T ), one gets u(f n )−z n ⇀ u(f λ ε )−z d ε in
T ) (3.18) Therefore, for allf ∈ F + , we deduce that
T ) = J λ ε (f λ ε ), (3.19) which means thatf λ ε ∈ F+ is the solution of the problem (3.5).
Next, we prove that the sequence {f n } n∈
L 2 (Q T ) By contradiction, suppose that the claim is false Then, we observe that n→∞lim ∥f n ∥ L 2 (Q
Therefore, there exists a (not relabeled) subsequence{f n } n∈
T ) = θ By choosing f = f λ ε ∈ F + in (3.19), we have
2θ 2 Combining with (3.20), we arrive at
T ), which contradicts with (3.18) The proof is finished.
On the other hand, regarding the error estimate of regularizing the source f + ∈ F + in Definition 3.1by the Tikhonov regularization, let us recall from
[29] the following result for the general linear inverse problems with convex constraints:
Lemma 3.6 states that if \( f^+ \) is the \( F^+ \)-best approximated source of the problem defined in (3.4), and \( f_\lambda^\epsilon \) is the solution to the regularized problem outlined in (3.5), then there exists an element \( \xi \in L^2(\omega_T) \) with the minimal \( L^2(\omega_T) \)-norm such that \( f^+ = \text{Proj}_{F^+}(A^* \xi) \) Here, \( A^* : L^2(\omega_T) \to L^2(Q_T) \) represents the adjoint operator of \( A \) in (3.4) Consequently, this leads to a significant inequality that is established in the context of this lemma.
Finite element discretization
The discrete regularized problem
We first discretize the regularized state and adjoint As in section 2.2.1.,let us define the discrete state problem: For ℓ ∈ L ∞ (QT) and f ∈ L 2 (QT), findu h (f) ∈ X h,0 that satisfies a h (u h (f), φ h ) = (ℓf, φ h ) L 2 (Q
The following discrete stability condition holds sup φ h ∈Y h \{0} a h (u h (f), φ h )
|||φ h ||| ≥ C|||u h (f)||| ∗ ∀u h (f) ∈ Xh,0, (3.22) which ensures that the problem (3.21) is uniquely solvable A priori estimates for the state error in three different norms have been presented in the previous chapter.
Similarly, we present the interface-fitted space-time method for solving the adjoint problem (3.9) We introduce the space
Xh,T = {φ h ∈ Yh | φh = 0onΩ× {T}} ⊂ XT The discrete adjoint problem reads as: Determinep h (f) ∈ X h,T such that a ′ h (p h (f), ϕ h ) = (χ ω T (u(f)−z d ε ), ϕ h ) L 2 (Q
T ) ∀ϕ h ∈ Y h , (3.23) with the bilinear forma ′ h : X T ×Y →Rgiven by a ′ h (p, ϕ) =− ⟨∂ t p, ϕ⟩+
Employing the technique as in (3.22), we establish the following stability condition sup ϕ h ∈Y h \{0} a ′ h (ph(f), ϕh)
The inequality \( |||ϕ h||| ≥ C|||p h (f)||| \) for all \( p h (f) \in X h,T \) establishes the unique solvability of the problem defined in equation (3.23) Additionally, by applying the principles used for the state problem, one can obtain a priori error estimates for the adjoint For clarity, we will outline the primary results needed in this context.
) , whereγ 0 > 0is a sufficiently large number, p(f) ∈ X T and p h (f) ∈ X h,T be the solutions of the problems (3.9) and (3.23), respectively Assume that there existsy ′ ∈ V0 andz ′ ∈ Xthat satisfy
(p(f)−p h (f))φdxdt ∀φ∈ Y (3.26) with z ′ (ã,0) ∈ H 1 0 (Ω) We have y ′ , z ′ ∈ H 1 (Q T ) ∩ H s (Q 1 ∪Q 2 ) with s > d+3
2 Furthermore, assume that there exists a constantC > 0 independent of p(f)and p h (f) such that
Lemma 3.7 Forf ∈ F + , let p(f) ∈ X T and p h (f) ∈ X h,T be the solutions of the problems(3.9)and (3.23), respectively. a) Assume that Assumption 2.1 is satisfied Then, we have the following estimate
1 ∪Q 2 ) (3.29) b) Moreover, if the problem (3.25) has a solution y ′ ∈ V 0 that satisfies (3.27), then there holds the estimate
1 ∪Q 2 ). c) Furthermore, if there exists a solution z ′ ∈ X of the problem(3.26) that satisfies (3.28), then the following estimate holds
The final step involves discretizing the regularized source using a variational approach This method transforms the discretization process into a discrete treatment for a term associated with the regularized adjoint Consequently, the discrete inverse source problem is formulated as minimizing \( f_h \) within the feasible set \( F \).
T ∈ L 2 (ω T ) denotes the discrete data Here, u ∗ h ∈
The function \( u_h \) approximates \( u^* \) in the interface-fitted space-time method, defined within the space \( X_{h,0} \) For the case of \( f_h \) in \( F^+ \) and \( \delta f_h \) in \( L^2(Q_T) \), the expression \( u_h(\delta f_h) = u_h(f_h + \delta f_h) - u_h(f_h) \) remains in \( X_{h,0} \) and serves as the solution to the problem characterized by \( a_h(u_h(\delta f_h), \phi_h) = (\ell \delta f_h, \phi_h)_{L^2(Q)} \).
Furthermore, the technique as in (3.16) gives us u h (δf h ), u h (f h )−z d,h ε
T ), (3.33) where u h (f h ) ∈ X h,0 and p h (f h ) ∈ X h,T be the solutions of the problems (3.21) and (3.23) with the corresponding right-hand sides ℓf h ∈ L 2 (Q T ) and χω T uh(fh)−z d,h ε
∈ L 2 (QT) By employing this equality, we can prove the following discrete optimality conditions:
Lemma 3.8 Let f λ,h ε ∈ F+ be the solution of the problem (3.31), u ε λ,h ∈
X h,0 and p ε λ,h ∈ X h,T denote the corresponding state and adjoint Then, the following optimality system is satisfied a h u ε λ,h , φ h
Error and convergence estimates
In this section, we analyze the error between the F + -best approximated solution f + and the solution f λ,h ε of the problem (3.31) in the L 2 (Q T )-norm We establish a relationship between this error and the parameters λ, mesh size h, and noise level ε Furthermore, we propose an a priori selection for λ based on h and ε, ensuring that f λ,h ε strongly converges to f + in L 2 (Q T ) as λ approaches 0 We begin our discussion with the application of the triangle inequality to the error term f + − f λ,h ε.
In equation (3.37), the initial term on the right side is addressed in Lemma 3.6 Our objective is to estimate the second error term in (3.37) and subsequently establish a priori criteria for the parameter λ, which guarantees the desired convergence based on the overall error.
The main result of this subsection is stated in Theorem3.12and Corollary 3.13 To begin, we denote by u h (f λ ε ) ∈ X h,0 andp h (f λ ε ) ∈ X h,T the solutions of the following problems a h (u h (f λ ε ), φ h ) = (ℓf λ ε , φ h ) L 2 (Q
Lemma 3.9 Let the triples (u h (f λ ε ), p h (f λ ε ), f λ ε ) ∈ X h,0 ×X h,T ×F + and u ε λ,h , p ε λ,h , f λ,h ε
∈ Xh,0×Xh,T ×F + be the solutions of the problems (3.38)- (3.39),(3.12) and (3.34)-(3.36), respectively Then, it holds that u h (f λ ε )−u ε λ,h
Proof First, let us prove the first inequality By subtracting (3.34) from (3.38), we obtain a h u h (f λ ε )−u ε λ,h , φ h
The stability condition (3.22) gives us
On the other hand, from the inequality (1.1), one has sup φ h ∈Y h \{0} a h u h (f λ ε )−u ε λ,h , φ h
We continue by subtracting (3.35) from (3.39) to get a ′ h p h (f λ ε )−p ε λ,h , ϕ h
Invoke the inequality (3.24), the technique as in (3.40), and the inequality (3.40) itself, one obtains p h (f λ ε )−p ε λ,h
L 2 (Q T ). (3.41) The first inequality follows by combining (3.40) and (3.41) The second one is a consequence of the first one, thanks to (1.1).
Lemma 3.10 Let (p ε λ , f λ ε ) ∈ X T ×F + and p h (f λ ε ) ∈ X h,T be the solutions of the problems (3.11)-(3.12) and (3.39), respectively Let f λ,h ε ∈ F + be the solution of the problem(3.36)in case of variational discretization Then, the following estimate holds f λ ε −f λ,h ε
Proof We choose f = f λ,h ε ∈ F + in (3.12) andf h = f λ ε ∈ F + in (3.36), then add the corresponding inequalities to get λ f λ ε −f λ,h ε
By using the Cauchy inequality, we estimate the first term on the right-hand side of (3.42) as follows
To handle the second term on the right-hand side of (3.42), we utilize the equality (3.33) twice We have
By combining (3.42), (3.43), and (3.44), we obtain the result.
We next estimate the right-hand side of the inequality in lemma 3.10 In doing so, let us introducepe h (f λ ε ) ∈ X h,T as the solution of the problem a ′ h (pe h (f λ ε ), ϕ h ) = (χ ω T (u ε λ −z d ε ), ϕ h ) L 2 (Q
Lemma 3.11 Let (u ε λ , p ε λ , f λ ε ) ∈ X 0 ×X T ×F + and p h (f λ ε ) ∈ X h,T be the solutions of the problems(3.10)-(3.12) and (3.39), respectively Let u ∗ ∈ X0 be the solution of the problem (3.2) Assume that the assumptions of Lemma 2.11and Lemma3.7c are satisfied Then, there holds the following
Proof We first derive the estimate (3.46) By using the triangle inequality, one has
|||p ε λ −p h (f λ ε )||| ∗ ≤ |||p ε λ −pe h (f λ ε )||| ∗ +|||pe h (f λ ε )−p h (f λ ε )||| ∗ , (3.48) where pe h (f λ ε ) ∈ X h,T is the solution of the problem (3.45) A priori error estimate (3.29) yields
1 ∪Q 2 ) (3.49) For dealing with the second term on the right-hand side of (3.48), we subtract (3.39) from (3.45) to get a ′ h (pe h (f λ ε )−p h (f λ ε ), ϕ h ) = χω T u ε λ −uh(f λ ε )−z d ε +z d,h ε
We invoke the technique as in (3.41) and a priori error estimate (2.24) to arrive at
The estimate (3.46) follows by substituting (3.49) and (3.51) into (3.48) We employ the same arguments to prove the estimate (3.47) Indeed, we have
By a priori error estimate (3.30), it holds
On the other hand, we apply the technique as in (3.41), and the inequality (3.51) to obtain
We arrive at the first main result of this section by combining Lemmas3.9, 3.10, and3.11with the triangle inequality.
The solutions to the problems outlined in equations (3.10)-(3.12) and (3.34)-(3.36) under variational discretization are denoted as X h,0 × X h,T × F Let u ∗ ∈ X 0 represent the solution to problem (3.2) Provided that the conditions of Lemma 2.11 and Lemma 3.7c are fulfilled, we can derive the estimates for u ε λ − u ε λ,h.
(3.53) Proof Let us sketch the proof of the first estimate Thanks to the triangle inequality, a priori error estimate (2.24), Lemmas 3.9and3.10, we have u ε λ −u ε λ,h
The conclusion follows from the estimate (3.46) The inequality (3.53) is proved similarly.
Finally, from the inequality (3.37), Lemma 3.6, and the inequality (3.53) in the previous theorem, we have the following result:
Corollary 3.13 Let f + ∈ F + be the F + -best approximated solution of the problem (3.4) and f λ,h ε ∈ F + be the solution of the problem (3.31) Assume that the assumptions of Theorem2.11, Lemma3.6and Lemma3.7c are satis- fied There holds the following estimate f + −f λ,h ε
Moreover, ifλ = O h 4 3 +ε then f λ,h ε → f + inL 2 (Q T ) as λ → 0with the convergence rateO h 2 3 +ε 1 2
We presented the interface-fitted space-time method for the advection- diffusion equation with a moving interface We showed two optimal order a priori error estimates under some appropriate conditions.
We derived error and convergence estimates for an inverse source problem related to an advection-diffusion scenario with moving subdomains Utilizing the interface-fitted space-time method, we addressed the regularized state and adjoint, while the regularized source was discretized through a variational approach Our findings include optimal order error estimates for the regularized source, state, and adjoint in two norms Additionally, we propose a priori choices for λ, ensuring that f λ,h ε converges to f + in L²(QT) as λ approaches 0, leading to the derivation of the convergence rate.
In future work, we aim to expand our findings to three-dimensional space Additionally, we plan to establish a priori error estimates that incorporate the parameter λ in the numerators of the fractions on the right-hand side, similar to the approach in [35], to achieve a higher convergence rate as indicated in corollary 3.13.
[1] S Gross and A Reusken Numerical Methods for Two-Phase Incom- pressible Flows Springer Berlin Heidelberg, 2011.
[2] M Slodiˇcka Parabolic problem for moving/ evolving body with perfect contact to neighborhood J Comput Appl Math., 391:113461, 2021.
[3] V C Le, M Slodiˇcka, and K Van Bockstal A time discrete scheme for an electromagnetic contact problem with moving conductor Applied Mathematics and Computation, 404:125997, 2021.
[4] V C Le, M Slodiˇ cka, and K Van Bockstal A numerical scheme for solving an induction heating problem with moving non-magnetic conductor Comput Math Appl., 171:60–81, 2024.
[5] Q H Nguyen, V C Le, P C Hoang, and T T M Ta A fitted space-time finite element method for parabolic problems with moving interfaces. Submitted for publication, 2024.
[6] I Babuˇska The finite element method for elliptic equations with dis- continuous coefficients Computing, 5(3):207–213, 1970.
[7] T Belytschko and T Black Elastic crack growth in finite ele- ments with minimal remeshing Internat J Numer Methods Engrg., 45(5):601–620, 1999.
[8] R Guo Solving parabolic moving interface problems with dynamical immersed spaces on unfitted meshes: fully discrete analysis SIAM J. Numer Anal., 59(2):797–828, 2021.
[9] C.-C Chu, I Graham, and T.-Y Hou A new multiscale finite ele- ment method for high-contrast elliptic interface problems.Math Comp., 79(272):1915–1955, 2010.
[10] P Zunino Analysis of backward Euler/extended finite element dis- cretization of parabolic problems with moving interfaces Comput. Methods Appl Mech Engrg., 258:152–165, 2013.
[11] C Lehrenfeld and A Reusken Analysis of a Nitsche XFEM-DG dis- cretization for a class of two-phase mass transport problems SIAM J. Numer Anal., 51(2):958–983, 2013.
[12] S Badia, H Dilip, and F Verdugo Space-time unfitted finite element methods for time-dependent problems on moving domains Comput. Math Appl., 135:60–76, 2023.
[13] C W Hirt, A A Amsden, and J L Cook An arbitrary Lagrangian- Eulerian computing method for all flow speeds J Comput Phys., 14(3):227–253, 1974.
[14] A M Winslow Numerical solution of the quasilinear Poisson equation in a nonuniform triangle mesh J Comput Phys., 1(2):149–172, 1966.
[15] M Bellassoued and M Yamamoto Inverse source problem for a trans- mission problem for a parabolic equation J Inverse Ill-Posed Probl., 14(1):47–56, 2006.
[16] S Chen, D Jiang, and H Wang Simultaneous identification of ini- tial value and source strength in a transmission problem for a parabolic equation Adv Comput Math., 48(6):Paper No 77, 28, 2022.
[17] Z Zhang, D Liang, and Q Wang Immersed finite element method and its analysis for parabolic optimal control problems with interfaces Appl. Numer Math., 147:174–195, 2020.
[18] M Hinze A variational discretization concept in control constrained op- timization: the linear-quadratic case Comput Optim Appl., 30(1):45–
[19] Dinh Nho H`ao, Le Thi Thu Giang, and Nguyen Thi Ngoc Oanh De- termination of the right-hand side in elliptic equations Optimization, 73(4):1195–1227, 2024.
[20] A Ern and J.-L Guermond Finite Elements I: Approximation and In- terpolation Springer International Publishing, 2021.
[21] Z Wu, J Yin, and C Wang Elliptic and Parabolic Equations World Scientific Publishing, 2006.
[22] F Tr¨oltzsch Optimal Control of Partial Differential Equations Ameri- can Mathematical Society, 2010.
[23] E M Stein Singular Integrals and Differentiability Properties of Func- tions Princeton University Press, 1971.
[24] J.-L Lions Sur les probl`emes mixtes pour certains syst`emes paraboliques dans des ouverts non cylindriques Ann Inst Fourier (Grenoble), 7:143–182, 1957.
[25] I Voulis and A Reusken A time dependent Stokes interface problem: well-posedness and space-time finite element discretization ESAIM: Math Model Numer Anal., 52(6):2187–2213, 2018.
[26] A Ern and J.-L Guermond Finite Elements II: Galerkin Approxima- tion, Elliptic and Mixed PDEs Springer International Publishing, 2021.
[27] Z Chen and J Zou Finite element methods and their convergence for elliptic and parabolic interface problems.Numer Math., 79(2):175–202, 1998.
[28] M Feistauer On the finite element approximation of a cascade flow problem Numer Math., 50(6):655–684, 1987.
[29] H W Engl, M Hanke, and A Neubauer Regularization of Inverse Problems Kluwer Academic Publishers, 1996.
[30] V K Ivanov, V V Vasin, and V P Tanana Theory of Linear Ill-Posed Problems and Its Applications VSP, 2002.
[31] B Hofmann and R Plato On ill-posedness concepts, stable solvability and saturation J Inverse Ill-Posed Probl., 26(2):287–297, 2018.