On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. II.Convergence of the Method of Finite Differences

We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consist of the minimization of the sum of $L_2$-norm declinations from the available measurement of the temperature flux on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. In {\it Inverse Problems and Imaging, 7, 2(2013), 307-340} we proved well-posedness in Sobolev spaces framework and convergence of time-discretized optimal control problems. In this paper we perform full discretization and prove convergence of the discrete optimal control problems to the original problem both with respect to cost functional and control.


Introduction and Motivation
Consider the general one-phase Stefan problem ( [15,26]): find the temperature function u(x, t) and the free boundary x = s(t) from the following conditions a(s(t), t)u x (s(t), t) + γ(s(t), t)s ′ (t) = χ(s(t), t), where a, b, c, f , φ, g, γ, χ, µ are known functions and a(x, t) ≥ a 0 > 0, s 0 > 0 (1. 6) In the physical context, f characterizes the density of the sources, φ is the initial temperature, g is the heat flux on the fixed boundary and µ is the phase transition temperature. Assume now that some of the data is not available, or involves some measurement error. For example, assume that the heat flux g(t) on the fixed boundary x = 0 is not known and must be found along with the temperature u(x, t) and the free boundary s(t). In order to do that, some additional information is needed. Assume that this additional information is given in the form of the temperature measurement along the boundary x = 0: u(0, t) = ν(t), for 0 ≤ t ≤ T (1.7) Inverse Stefan Problem (ISP): Find the functions u(x, t) and s(t) and the boundary heat flux g(t) satisfying conditions (1.1)-(1.7). Motivation for this type of inverse problem arose, in particular, in the modeling of bioengineering problems on the laser ablation of biological tissues through Stefan problem (1.1)-(1.6), where s(t) is the ablation depth at the moment t. The boundary temperature measurement u(0, t) contains an error, which makes it impossible to get reliable measurement of the boundary heat flux g(t), and the ISP must be solved for its identification. This approach allows us to regularize an error contained in a measurement ν(t). Another advantage of this approach is that, in fact, condition (1.5) can be treated as a measurement of the temperature on the ablation front, and our approach allows us to regularize an error contained in temperature measurement µ(t) on the ablation front. Still another important motivation arises in optimal control of the Stefan problem, where controlling g(t) is equivalent of controlling external temperature along the fixed boundary. It should be pointed out that the method of this paper can be applied to different type of inverse problems. For example, (1.7) can be replaced with u(x, T ) = w(x), for 0 ≤ x ≤ s(T ), meaning that measurements are taken for the final temperature distribution w(x) and final ablation depth s(T ). Instead of identification of the boundary flux g, one can consider the inverse free boundary problem with any of the unknown coefficients a, b, c or right hand side f . The ISP is not well posed in the sense of Hadamard. If there is no coordination between the input data, the exact solution may not exist. Even if it exists, it might be not unique, and most importantly, there is no continuous dependence of the solution on the data. The ISP was first mentioned in [10], in the form of finding a heat flux on the fixed boundary which provides a desired free boundary. This problem is similar to a non-characteristic Cauchy problem for the heat equation. The variational approach for solving this ill-posed inverse Stefan problem was performed in [7,8]. The first result on the optimal control of the Stefan problem appeared in [36]. It consists of finding the optimal value of the external temperature along the fixed boundary, in order to ensure that the solutions of the Stefan problem are close to the measurements taken at the final moment. In [36], the existence result was proved. In [38] the Frechet differentiability and the convergence of the difference schemes was proved for the same problem and Tikhonov regularization was suggested. Later development of the inverse Stefan problem was along these two lines: Inverse Stefan problems with given phase boundaries were considered in [2,4,6,9,11,12,13,18,32,16]; optimal control of Stefan problems, or equivalently inverse problems with unknown phase boundaries were investigated in [3,14,19,20,21,22,23,25,29,27,30,31,35,16]. We refer to monography [16] for a complete list of references of both types of inverse Stefan problems, both for linear and quasilinear parabolic equations. The main methods used to solve the inverse Stefan problem are based on variational formulation, method of quasi-solutions or Tikhonov regularization which takes into account ill-posedness in terms of the dependence of the solution on the inaccuracy involved in the measurement (1.7), Frechet differentiability and iterative conjugate gradient methods for numerical solution. Despite its effectiveness, this approach has some deficiencies in many practical applications: • Solution of the inverse Stefan problem is not continuously dependent on the phase transition temperature µ(t): small perturbation of the phase transition temperature may imply significant change of the solution to the inverse Stefan problem. Accordingly, any regularization which equally takes into account instability with respect to both ν(t) from measurement (1.7), and the phase transition temperature µ(t) from (1.5) will be preferred. It should be also mentioned that in many applications the phase transition temperature is not known explicitly. In many processes the melting temperature of pure material at a given external action depends on the process evolution. For example, gallium (Ga, atomic number 31) may remain in the liquid phase at temperatures well below its mean melting temperature ( [26]).
• Numerical implementation of iterative gradient type methods within the existing approach requires solving the full free boundary problem at every step of the iteration, and accordingly has quite a high computational cost. An iterative gradient method which requires solution of the boundary value problem in a fixed region at every step would definitely be much more effective in terms of the computational cost.
The main goal of this project is to develop a new variational approach based on the optimal control theory which is capable of addressing both of the mentioned issues and allows the inverse Stefan problem to be solved numerically with least computational cost by using conjugate gradient methods in Hilbert spaces. In [1] we proved the existence of the optimal control and convergence of the family of time-discretized optimal control problems to the continuous problem. In this paper we perform full discretization through finite differences and prove the convergence of the discrete optimal control problems to the continuous problem both with respect to cost functional and control. We employ Sobolev spaces framework which allows us to reduce the regularity and structural requirements on the data. We address the problems of Frechet differentiability and application of iterative conjugate gradient methods in Hilbert spaces in an upcoming paper.
Throughout the paper we use the usual notation for Sobolev spaces according to references [24,5,28,33,34]. Notation is described below in Section 1.2.

Notation of Sobolev Spaces
T ] whose weak derivatives up to order k belongs to L 2 [0, T ] and scalar product is defined as (Ω) -Hilbert space of all elements of L 2 (Ω) whose weak derivative ∂u ∂x belongs to L 2 (Ω), and scalar product is defined as (Ω) -Hilbert space of all elements of L 2 (Ω) whose weak derivatives ∂u ∂x , ∂u ∂t belong to L 2 (Ω), and scalar product is defined as (Ω) -Banach space which is the completion of W 1,1 2 (Ω) in the norm of V 2 (Ω). It consists of all elements of V 2 (Ω), continuous with respect to t in norm of L 2 [0, s(t)] and with finite norm (Ω) -Hilbert space of all elements of L 2 (Ω) whose weak derivatives ∂u ∂x , ∂u ∂t , ∂ 2 u ∂x 2 belong to L 2 (Ω), and scalar product is defined as

Optimal Control Problem
Consider a minimization of the cost functional on the control set for arbitrary Φ ∈ W 1,1 2 (Ω) We also need a notion of weak solution from V 2 (Ω) of the Neumann problem: for arbitrary Φ ∈ W 1,1 2 (Ω) such that Φ| t=T = 0. If u is a weak solution either from V 2 (Ω) (or W 1,1 2 (Ω)), then traces u| x=0 and u| x=s(t) are elements of L 2 [0, T ], when s ∈ W 2 2 [0, T ] ( [28,24]) and cost functional J (v) is well defined. Furthermore, formulated optimal control problem will be called Problem I.

Discrete Optimal Control Problem
Let ω τ = {t j = j · τ, j = 0, 1, . . . , n} be a grid on [0, T ] and τ = T n . Consider a discretized control set where we assign s −1 = s 0 and use the standard notation for the finite differences: Introduce two mappings Q n and P n between continuous and discrete control sets: (1.11) Let us now introduce spatial grid. Given [v] n ∈ V n R , let (p 0 , p 1 , · · · , p n ) be a permutation of (0, 1, · · · , n) according to order s p 0 ≤ s p 1 ≤ · · · ≤ s pn In particular, according to this permutation for arbitrary k there exists a unique j k such that Furthermore, unless it is necessary in the context, we are going to write simply j instead of subscript j k . Let n , · · · , N} of stepsize order h, i.e. h = O(h) as h → 0. Furthermore we simplify the notation and write m and assume that m k → +∞, as n → ∞.
Introduce Steklov averages where i = 0, 1, · · · , N − 1; k = 1, · · · , n; d stands for any of the functions a, b, c, f , and h stands for any of the functions ν, µ, g or g n . Given v = (s, g) ∈ V R we define Steklov averages of traces (1.14) ∈ V n R we define Steklov averages χ k s n and (γ s n (s n ) ′ ) k through (1.14) with s replaced by s n from (1.11).
Let φ n be a piecewise constant approximation of φ: Next we define a discrete state vector through discretization of the integral identity (1.9) (b) Recalling (1.12), for arbitrary k = 1, · · · , n first m j + 1 components of the vector u(k) ∈ R N +1 solve the following system of m j + 1 linear algebraic equations: (1.15) (c) For arbitrary k = 0, 1, ..., n, the remaining components of u(k) ∈ R N +1 are calculated as where [r] means integer part of the real number r.
It should be mentioned that for any k = 1, 2, · · · , n system (3.61) is equivalent to the following summation identity for arbitrary numbers η i , i = 0, 1, · · · , m j . Consider a discrete optimal control problem of minimization of the cost functional on a set V n R subject to the state vector defined in Definition 1.3. Furthermore, formulated discrete optimal control problem will be called Problem I n .
Throughout, we use piecewise constant and piecewise linear interpolations of the discrete state vector: given discrete state vector [u([v] n )] n = (u(0), u(1), ..., u(n)), let . As before, we employ standard notations for difference quotients of the discrete state vector:

Formulation of the Main Result
Throughout the whole paper we assume the following conditions are satisfied by the data: Our main theorems read: Note that Theorem 1.1 was already proved in [1] by using method of lines.
Theorem 1.2 Sequence of discrete optimal control problems I n approximates the optimal control problem I with respect to functional, i.e. where In particular s n converges to s * uniformly on [0, T ]. For any δ > 0, define

Preliminary Results
In Lemma 2.1 below we prove existence and uniqueness of the discrete state vector [u([v] n )] n (see Definition 1.3) for arbitrary discrete control vector [v] n ∈ V n R . In Lemma 2.2 we remind a general approximation criteria for the optimal control problems from ( [37]). In Lemma 2.3 we recall some properties of the mappings Q n and P n between continuous and discrete control sets.
Lemma 2.1 For sufficiently small time step τ , there exists a unique discrete state vector [u([v] n )] n for arbitrary discrete control vector [v] n ∈ V n R . Proof. As it is mentioned above, for any k = 1, 2, · · · , n system (3.61) is equivalent to the summation identity (1.17) for arbitrary numbers η i , i = 0, 1, · · · , m j . Let {ũ i (k)} be a solution of the homogeneous system related to (3.61), i.e.
Using (1.6) and Cauchy inequality with ǫ > 0 we derive that From (2.3) it follows thatũ i (k) = 0, i = 0, 1, · · · , m j , and hence the homogeneous system only has a trivial solution for τ < τ 0 . Accordingly, system is uniquely solvable and therefore, for any given discrete control vector [v] n there exists a unique discrete state vector defined by Definition 1.3. Lemma is proved. The following known criteria will be used in the proof of Theorem 1.2.

Lemma 2.2 [37]
Sequence of discrete optimal control problems I n approximates the continuous optimal control problem I if and only if the following conditions are satisfied: (1) for arbitrary sufficiently small ǫ > 0 there exists number and for any fixed ǫ > 0 and for all v ∈ V R−ǫ the following inequality is satisfied: (2.5) (3) the following inequalities are satisfied: Next lemma demonstrates that the mappings Q n and P n introduced in Section 1.4 satisfy the conditions of Lemma 2.2.
For arbitrary sufficiently small ǫ > 0 there exists n ǫ such that and n > n ǫ .
Let us know choose [v] n ∈ V n R . We simplify the notation and assume v = (s, g) = P n ([v] n ). Through direct calculations we derive where C is independent of τ . Furthermore, we use notation C for all (possibly different) constants which are independent of τ . By using CBS inequality we have By applying Morrey inequality to s ′ (t) from (2.15) it follows Therefore from (2.14),(2.16),(2.17) it follows that for sufficiently small τ In a similar way we calculate where C ′ is independent of n.

and by Morrey inequality
and hence for the first component Note that for the step size h i we have one of the three possibilities:

First Energy Estimate and its Consequences
The main goal of this section to prove the following energy estimation for the discrete state vector.
where C is independent of τ and 1 + be an indicator function of the positive semiaxis.
First we prove the following lemma.
where C is independent of τ .
Proof. By choosing η i = 2τ u i (k) in (1.17) and by using the equality Using (1.6), Cauchy inequalities with appropriately chosen ǫ > 0, and Morrey inequality max 0≤i≤m j where C * , C are independent of τ and [u([v] n )] n , from (3.3) we derive that where C 1 is independent of τ . Assuming that τ < C 1 , from (3.5) it follows that By induction we have For arbitrary 1 ≤ l ≤ k ≤ n we have as τ → 0. Accordingly for sufficiently small τ we have By applying Cauchy-Bunyakovski-Schwartz (CBS) inequality from (3.7)-(3.9) it follows that max 1≤k≤n where C 2 is independent of τ . Having (3.10), we perform summation of (3.5) with respect to k from 1 to n and derive From (3.10) and (3.11), (3.2) follows. Lemma is proved. Proof of Theorem 3.1: Due to (3.2), it is enough to show that the left hand side of (3.1) is bounded by the left hand side of (3.2). By using reflective continuation property ofû(x; k) we easily derive that By using (1.13) and (2.22) we have (3.14) Proof. In addition to quadratic interpolation of [s] n from (1.11), consider two linear interpolations: where C * is independent of n. Our next goal is to absorb the last term on the right hand side of (3.1) into the left hand side. We have Note that if s k+1 > s k , then all the factors h i in the second term are bounded by s k+1 − s k and by using (2.21) we have Due to reflective continuation we have

From (3.17) and (3.18) it follows that
Assuming that τ is sufficiently small and by using (3.16) -(3.19) in (3.1), we absorb the last term on the right hand side of (3.19) into the left hand side of (3.1) and derive modified (3.1) with a new constant C: We can now estimate the last term on the right hand side of (3.20) as in [1]: By applying CBS inequality we have From the results on traces of the elements of space V 2 (D) ( [24,5,28]) it follows that for arbitrary u ∈ V 2 (D) the following inequality is valid with the constantC being independent of u as well as n. From (3.15),(3.22) and (3.23) it follows that If the constant C * from (3.15) satisfies the condition then from (3.20) and (3.24) it follows that where C is another constant independent of n. By applying the results on the traces of elements of W 1,0 2 (D) ( [5,28]) on smooth curve x = s n (t), Morrey inequality for (s n ) ′ and where C 3 is independent of γ, χ and n. Hence, from (3.26), (3.27) it follows the estimation If (3.25) is not satisfied, then due to (2.21) we can partition [0, T ] into finitely many segments [t n j−1 , t n j ], j = 1, q with t n 0 = 0, t nq = T in such a way that by replacing [0, T ] with any of the subsegments [t n j−1 , t n j ] (3.15) will be satisfied with C * small enough to obey (3.25). Hence, we divide D into finitely many subsets is uniformly bounded through the right-hand side of (3.28). Summation with j = 1, . . . , q implies (3.28).
Since (φ n , g n ) converge to (φ, g) strongly in L 2 [0, s 0 ] × L 2 [0, T ], from (3.28) it follows that the sequence {u τ } is weakly precompact in W 1,0 2 (D). Let u ∈ W 1,0 2 (D) be a weak limit point of u τ in W 1,0 2 (D), and assume that whole sequence {u τ } converges to u weakly in W 1,0 2 (D). Let us prove that in fact u satisfies the integral identity (1.10) for arbitrary test function Φ ∈ W 1,1 2 (Ω) such that Φ| t=T = 0. Due to density of C 1 (Ω) in W 1,1 2 (Ω) it is enough to assume Φ ∈ C 1 (Ω). Without loss of generality we can also assume that Φ ∈ C 1 (D T +τ ), Φ ≡ 0, for T ≤ t ≤ T + τ , where D T +τ = {(x, t) : 0 < x < l + 1, 0 < t ≤ T + τ } Otherwise, we can continue Φ to D T +τ with the described properties. Let Obviously, the sequences {Φ τ }, {Φ τ x } and {Φ τ t } converge as τ → 0 uniformly in D to Φ, ∂Φ ∂x and ∂Φ ∂t respectively. By choosing in (1.17) η i = τ Φ i (k), after summation with respect to k = 1, n and transformation of the time difference term as follows First note that the sequence {ũ τ } is equivalent to the sequence {u τ } in strong, and accordingly also in a weak topology of L 2 (D), and hence converges to u weakly in L 2 (D). Indeed, by using (3.1) we have |∆| denotes the Lebesgue measure of ∆. Sinces n (t k ) = s k , we have and all of the integrands are uniformly bounded in L 1 (D), it follows that the first term in the expression of R converges to zero as τ → 0. In a similar way one can see that the second, third and fifth terms also converge to zero as τ → 0. The fourth term in the expression of R converges to zero due to Corollary 2.1 and uniform convergence of {Φ τ } in D. To prove the convergence to zero of the last term of R, let Since the integrands are uniformly bounded in L 2 (D), the expression in (3.32) converges to zero as τ → 0. Hence, we have lim Due to weak convergence of u τ to u in W 1,0 2 (D), weak convergence ofũ τ to u in L 2 (D) and uniform convergence of the sequences {Φ τ }, { ∂Φ τ ∂x } and {Φ τ t } to Φ, ∂Φ ∂x and ∂Φ ∂t respectively, passing to limit as τ → 0, it follows that first, second and fourth integrals on the left-hand side of (3.30) converge to similar integrals with u τ (orũ τ ), Φ τ , Φ τ t , Φ τ (x, τ ), g n (t), φ n (x) and Φ τ (0, t) replaced by u,Φ, ∂Φ ∂t , Φ(x, 0), g(t), φ(x) and Φ(0, t) respectively. Since s n converges to s strongly in W 1 2 [0, T ], the traces γ(s n (t), (t)), χ(s n (t), t) converge strongly in L 2 [0, T ] to traces γ(s(t), (t)), χ(s(t), t) respectively. Since Φ τ (s n (t), t) converge uniformly on [0, T ] to Φ(s(t), t), passing to the limit as τ → 0, the last integral on the left-hand side of (3.30) converge to similar integral with s n and Φ τ replaced by s and Φ.
It only remains to prove that lim τ →0 Since {u τ } converges to u weakly in W 1,0 2 (D), it follows that and replace the norm of φ n in the first energy estimate through norm of φ.

Second Energy Estimate and its Consequences
Let given discrete control vector [v] n , along with discrete state vector [u([v] n )] n , the vector is defined asũ The main goal of this section to prove the following energy estimation for the vectorũ([v] n ).

Theorem 3.3
For all sufficiently small τ discrete state vector [u([v] n )] n satisfies the following stability estimation:

37)
Proof: Note that if s k−1 ≥ s k then u i (k) can be replaced throughũ i (k) in all terms of (1.17). By choosing η i = 2τũ it (k) in (1.17) and by using the equality If s k−1 < s k , then u i (k) can be replaced throughũ i (k) in all but in the term including backward discrete time derivative in (1.17). The latter will be estimated with the help of the following inequality: To prove (3.40), we transform the left hand side with the help of the CBS and Cauchy inequalty with ǫ = τ to derive which implies (3.40) due to (2.21). Hence, in general (3.39) is replaced with the inequality By adding inequalities (3.42) with respect to k from 1 to arbitrary p ≤ n we derive By using (1.6) and by applying Cauchy inequalities with appropriately chosen ǫ > 0, from (3.43) it follows that a 0 where C is independent of n. First term on the right hand side will be estimated as follows: Due to arbitraricity of p, from (3.44) it follows then the first term on the right hand side of (3.46) is absorbed into the first term on the left hand side. If (3.47) is not satisfied, then we can partition [0, T ] into finitely many subsegments which obey (3.47), absorb first term on the right hand side into the left hand side in each subsegment and through summation achieve the same for (3.46) in general.
with some C independent of n.
Hence, by replacing in the original problem (1.1)-(1.4) u with u−w we can derive modified (3.48) without the last three terms on the right-hand side and with f , replaced by (3.54) By using the stability estimation (3.2), from modified (3.48),(3.53) and (3.54), the following estimation follows: By estimating the last term on the right hand side of (3.55) as in the proof of Theorem 3.2, (3.37) follows. Theorem is proved. Second energy estimate (3.37) allows to strengthen the result of Theorem 3.2.

Theorem 3.4 Let
[v] n ∈ V n R , n = 1, 2, ... be a sequence of discrete controls and the sequence and substitute to derive From which by using (1.13) it follows that where C is independent of n or τ . Our goal is to prove that the right hand side of (3.57) is bounded by the left hand side of (3.37) for sufficiently large n. It is sufficient to prove the following claim: for fixed ǫ m , there exists N = N(ǫ m ) such that for ∀n > N s m k < min(s k , s k−1 ), k = 1, . . . , n (3.58) Indeed from (3.58) it follows that To prove (3.58), we first show that for sufficiently large n and all t k−1 ≤ t ≤ t k We have Morrey's inequality and (2.1), Similarly, there exists some N 2 = N 2 (ǫ m ), such that for n > N 2 and for all t k−1 ≤ t ≤ t k , (3.60) is true with s k replaced by s k−1 . Hence, (3.58) follows with N = max(N 1 , N 2 ). Applying (3.57), (3.59), second energy estimate (3.37), and the first energy estimate (3.1), derive By estimating the last term on the right hand side of (3.62) as in the proof of Theorem 3.2, we derive Since φ n → φ strongly in L 2 [0, s 0 ], g n → g weakly in W 1 2 [0, T ], the right hand side is uniformly bounded independent of n. Hence, {û τ } is weakly precompact in W 1,1 2 (Ω m ). It follows that it is strongly precompact in L 2 (Ω m ). Let u be a weak limit point of {û τ } in W 1,1 2 (Ω m ), and therefore a strong limit point in L 2 (Ω m ). From anothe side the sequences {û τ } and {u τ } are equivalent in strong topology of L 2 (Ω m ). Indeed, we have for all n > N(m) due to second energy estimate (3.37). Therefore, u is a strong limit point of the sequence {u τ } in L 2 (Ω m ). By Theorem 3.2 whole sequence {u τ } converges weakly in W 1,0 2 (Ω) to the unique weak solution from V 1,0 2 (Ω) of the problem (1.1)-(1.4). Hence, u is a weak solution of the problem (1.1)-(1.4) and we conclude that whole sequence {û τ } converges weakly in W 1,1 2 (Ω m ) to u ∈ W 1,1 2 (Ω m ) which is a weak solution of the problem (1.1)-(1.4) from V 1,0 2 (Ω). Hence, u t exists in Ω m and u t L 2 (Ωm) is uniformly bounded by the right hand side of (3.63). It easily follows that the weak derivative u t exists in Ω, and u ∈ W 1,1 2 (Ω). By using wellknown property of the weak convergence, and passing to limit first as n → +∞, and then as m → +∞, from (3.63), (3.56) follows. Theorem is proved.
In particular, Theorem 3.4 implies the following existence result: Corollary 3.2 For arbitrary v = (s, g) ∈ V R there exists a weak solution u ∈ W 1,1 2 (Ω) of the problem (1.1)-(1.4) which satisfy the energy estimate (3.56). By Sobolev extension theorem u can be continued to W 1,1 2 (D) with the norm preservation: Remark: In fact, we proved slightly higher regularity of u, and both in Theorem 3.4 and Corollary 3.2 W 1,1 2 (Ω) or W 1,1 2 (D)-norm on the left-hand sides of (3.56) or (3.65) can be replaced with The proof of the Theorem 1.1 coincides with the proof of identical Theorem 1.1 in [1]. The main idea is that first and second energy estimates imply weak continuity of the functional Since V R is weakly compact existence of the optimal control follows from Weierstrass theorem in weak topology.
We split the remainder of the proof of Theorem 1.2 into three lemmas.
The proof of Lemma 3.2 coincides with the proof of identical lemma 3.9 from [1]. [v] n )] n be a corresponding discrete state vector. In Theorem 3.4 it is proved that the sequence {û τ } converges to u weakly in W 1,1 2 (Ω m ) for any fixed m. This implies that the sequences of traces {û τ (0, t)} and {û τ (s(t) − ǫ m , t)} converge strongly in L 2 [0, T ] to corresponding traces u(0, t) and u(s(t) − ǫ m , t). Let us prove that that the sequences of traces {u τ (0, t)} and {u τ (s(t)−ǫ m , t)} converge strongly in L 2 [0, T ] to traces u(0, t) and u(s(t) − ǫ m , t) respectively. By Sobolev embedding theorem ( [5,28]) it is enough to prove that the sequences {u τ } and {û τ } are equivalent in strong topology of W 1,0 2 (Ω m ). In Theorem 3.4 it is proved that they are equivalent in strong topology of L 2 (Ω m ). It remains only to demonstrate that the sequences of derivatives ∂u τ ∂x and ∂û τ ∂x are equivalent in strong topology of L 2 (Ω m ). Following the proof of the Theorem 3.4, from the second energy estimate (3.37) it follows that for all n > N(m) Estimate the first term on the right-hand side of (3.76) as