NUMERICAL APPROXIMATION OF THE PHASE-FIELD TRANSITION SYSTEM WITH NON-HOMOGENEOUS CAUCHY-NEUMANN BOUNDARY CONDITIONS IN BOTH UNKNOWN FUNCTIONS VIA FRACTIONAL STEPS METHOD (cid:3)

The paper concerns with the proof of the convergence for an iterative scheme of fractional steps type associated to the phase-(cid:12)eld transition system endowed with non-homogeneous Cauchy-Neumann boundary conditions, in both unknown functions. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of nonlinear parabolic system. On the basis of this approach, a numerical algorithm in 2D case is introduced and an industrial implementation is made.

In the formulation of problem (1.1) we have started from the phase field equations describing the phenomenon of solidification (see [11]) to which we have added some new physics parameters, as well as appropriate boundary conditions, in order to cover a wide variety of industrial applications (see [18][19][20]).
The non-homogeneous Cauchy-Neumann boundary conditions in both unknown functions u and φ (see relation (1.2)), untreated until now in mathematical literature, include a broad class of complex phenomenas at ∂Ω and will thus allow the formulation of new boundary optimal control problems.
At the moment t the material is considered to be liquid if the phase function φ is close to +1 + δ 1 and u(t, x) ≥ 1 + δ 2 , while it is considered to be solid if the phase function φ is close to −1 − δ 1 and u(t, x) ≤ −1 − δ 2 , with δ 1 , δ 2 prescribed positive numbers.
We define the separating region (the interface at the moment t) as being the set: Regarding the existence and regularity of solutions in (1.1)-(1.3),we have where the constant C depends on |Ω| (the measure of Ω), T, n, p, q and physical parameters.Moreover, given any number M > 0, if (u 1 , φ 1 ) and (u 2 , φ 2 ) are solutions of (1.1)- (1.3) for the same initial conditions, corresponding to the dates where the constant C depends on |Ω|, T, M, n, p, q and physical parameters.

The symbol ∫
Q above denotes the duality between L 2 ([0, T ]; H 1 (Ω)) and L 2 ([0, T ]; which is relevant for industrial applications.Such a choice can be made also in the case n = 3, p > 5  2 (see also [9]).Throughout this paper when is not clearly precised, we will denote by C a constant which may change from line to line.
This paper is divided as follows: we start by giving the convergence of the linear approximating scheme (1.7)-(1.9)associated to the nonlinear transition system (1.1)-( 1.3) and finish by a numerical algorithm in the 2D case and industrial implementation.

Convergence and weak stability of the approximating scheme
In this section, we will prove the convergence of the iterative scheme (1.7)-(1.10) of fractional steps type for the phase-field transition system (1.1)-(1.3).We have ) be the solution of the approximating scheme (1.7)-(1.9).Then for ε → 0, one has (2.1) is the weak solution to the nonlinear phase transition system (1.1)-(1.3).
The following lemmas, which targets the Cauchy problem (1.10) and which are very useful in the proof of the main result of this Section (Theorem 2.1) were established for the first time in the work [16].For reader convenience we fully reproduce their proofs.
From the Cauchy problem (1.10), using the method of separation of variables and integrating on (iε, (i + 1)ε), we get This gives us Proof.The proof follows directly to Lemma 2.1.In fact, using (2.3) and relation (1.9) 2 , we deduce as claimed.
As regards the unknwown functions u * (t), φ * (t), one can prove as in [6] that it is absolutely continuous in t on [0, T ] and satisfies a.e the phase-field system (1.1)-(1.3),which means that the pair (u * (t), φ * (t)) is a strong solution to our nonlinear problem.So, Theorem 2.1 can be regarding as a constructive way to prove the existence and regularity of solutions for the nonlinear parabolic system (1.1)- (1.3).

Approximation of phase-field transition system in 2D by finite element method Algorithm Armel-fracfem2D
In this Section we are concerned with the numerical approximation of the weak solution corresponding with (1.7)-(1.9)(see Definition 1.2) by finite element method (fem) i.e. with the numerical approximation of the weak solution of the following equations: ) together with the initial conditions u(0, x) = u 0 (x), φ(0, x) = φ 0 (x), x ∈ Ω. Considering M = M ε as the number of equidistant nodes in which is divided the time-interval [0, T ], we set We assume that Ω ⊂ R 2 is a polygonal domain.Let T ρ be the triangulation (mesh) over Ω and Ω = ∪{K, K ∈ T ρ } and let N j = (x k , y l ), j = 1, nn, be the nodes associated to T ρ .If we denote by V nn the corresponding finite element space to T ρ then, the basis functions {b j } nn j=1 of V nn are defined by and We say that the function v(x, y) belongs to V nn only if it can be expressed as For i = 1, M , we denote by u i and φ i the V nn interpolant of u ε and φ ε , respectively.Then u i , φ i ∈ V nn and ) where the unknowns Let now U, Φ ∈ V nn be two arbitrary functions, i.e.
Using an implicit (backward) finite difference scheme in time and taking into account the above notations, we introduce the discrete equations corresponding to (3.1)-(3.2) as follows (i = 1, M ) for i = 1, M , where P4. (V nn ) A discrete analogous of (V);

P5.
Assemble the linear system of equations; P6.
Solve the system obtained in P5.
Corresponding to the physical parameters of the mathematical model (1.1)-(1.3),we have used industrial values indicated in the works [19,20].To generate the triangulation T ρ , we consider Ω a cross-section in a slab (thin) of 1300mm×220mm.In Figure 1, the mesh can be seen in the directions of x 1 and x 2 -axis of a rectangular profile.The initial solution u 0 l , φ 0 l in (3.13) was computed as solution of stationary equation φ t = ∆φ = 0 and as solution of Cauchy problem (1.10), respectively.
The values of w 1 (t, x) and w 2 (t, x) ∈ Σ are given as a spline interpolation (only the mobile part of the continuous casting machine is illustrated in Figures 2 and 3    The shape of the graphs shows the numerical stability and accuracy of the results obtained by implementing the fractional steps method (1.7) and (1.10).The most interesting aspect that we can observe while analyzing Figures 4-7 is the presence of supercooling and superheating phenomena (presence of solid fractions in the liquid, for example).
The numerical solution computed by this way can be considered as an admissible one for the corresponding boundary optimal control problem in order to improve the process optimization of continuous casting.Generally the fractional steps method considered here can used to approximate the solution of a nonlinear parabolic phasefield system containing a general nonlinear part.

Figure 2 .
Figure 2. The values w 1 (t, x) on the mobile part

Figure 3 .
Figure 3.The values w 2 (t, x) on the mobile part

Figures 4 and 6
Figures 4 and 6 represent the approximate solutions u i (i = 2 and i = M = 8, respectively), while Figures 5 and 7 represent the approximate solutions φ i (i = 2 and i = M = 8, respectively). ds.