Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions

The paper concerns with the existence, uniqueness, regularity and the approximation of solutions 
to the nonlinear phase-field (Allen-Cahn) equation, endowed with 
non-homogeneous dynamic boundary conditions (depending both on time and space variables). 
It extends the already studied types of boundary conditions, 
which makes the problem to be more able to describe many important phenomena of two-phase systems, 
in particular, the interactions with the walls in confined systems. 
The convergence and error estimate results for an iterative scheme of fractional steps type, 
associated to the nonlinear parabolic equation, are also established. 
The advantage of such method consists in simplifying the numerical computation. 
On the basis of this approach, a conceptual numerical algorithm is formulated in the end.


1.
Introduction. Let us consider the following nonlinear parabolic boundary value problem with respect to the unknown function ϕ where: • Ω denotes some bounded domain in IR n (1 < n ≤ 3) with a C 2 boundary ∂Ω = Γ. We set Q = (0, T ] × Ω, Σ = (0, T ] × ∂Ω, where T > 0 stands for some final time. Of course, t ∈ [0, T ] while x varies in Ω; • ϕ(t, x) is the phase function (the order parameter), used to distinguish between the states (phases) of material which occupies the region Ω at every time t ∈ [0, T ]; • α, ξ, c 0 are physical parameters representing: the relaxation time, the measure of the interface thickness and a positive constant, respectively;

ALAIN MIRANVILLE AND COSTICȂ MOROŞANU
• ∆ Γ is the Laplace-Beltrami operator; • g(t, x) ∈ L p (Q) is a given function (can be interpreted as distributed control), where p satisfies p ≥ n + 2 2 ; (2) • w(t, x) ∈ W 1− 1 2p ,2− 1 p p (Σ) is a given function (can be interpreted as boundary control); . For the remaining data in (1), we keep the meanings already formulated above. Consequently, the nonlinear parabolic boundary value problem (1) can be rewritten suitably in the following form We first analyze the linearized version of problem (4), that is which are useful in the proof of the main result of this Section. We have , then problem (5) possesses a unique solution (ϕ, ψ) ∈ W 1,2 p (Q) × W 1,2 p (Σ) such that ϕ W 1,2 where C depends on |Ω|, T , n, p, α, ξ, c 0 , but is independent of ϕ, ψ,f and h 3 .
Proof. Applying Lemma 2.3 in [26] with h 3 = −w and making use of the embeddings (2)), we can easily conclude that the results set out by Lemma 2.1 are true.
The main result of this Section establishes the dependence of the solution (ϕ, ψ) in the nonlinear parabolic problem (4) on the terms g(t, x) and w(t, x). We have where the constant C depends on |Ω|, T , n, p, α, ξ and c 0 but is independent of ϕ, ψ, g and w.
2.1. Proof of Theorem 2.2. Here we will prove the existence, uniqueness and regularity of the solution to problem (4), considering as nonlinear term the classical regular potential f (ϕ) = 1 2ξ (ϕ − ϕ 3 ) which verifies for n ≤ 3 and r = 3 the general assumptions H 1 and H 3 formulated in [32], namely: H 3 : There are a functionF : IR 2 → IR and a constant b 0 > 0 verifying the relations: Basic tools in treating the problem (4) are the Leray-Schauder degree theory [14], the L p -theory of linear and quasi-linear parabolic equations [24], as well as the Lions and Peetre embedding Theorem [25], p. 24, which ensures the existence of a continuous embedding W 1,2 p (Q) ⊂ L µ (Q), where the number µ is defined as follows In order to use the Leray-Schauder degree theory, as we have mentioned above, we will choose as suitable Banach space B = L 3p (Q) × L p (Σ), endowed with the norm Let us define the nonlinear operator T : where (ϕ, ψ) is the solution to the linear problem The nonlinear operator T defined by (12), enjoys the following properties: A. T is well-defined (problem (13) has a solution). We remember that g ∈ L p (Q) and w(t, x) ∈ W 1− 1 2p ,2− 1 p p (Σ) are given functions. It follows from the right hand of (13) 1 that ∀v 1 ∈ L 3p (Q), we deduce that 1 x) ∈ L p (Q). Applying Lemma 2.1 with the settings: the solution (ϕ, ψ) to problem (13) exists and is unique. Furthermore, ∀v ∈ B and ∀ λ ∈ [0, 1], Since µ = p (n + 2) > 0, we can take µ > 3p in all cases required by (10). Consequently, we have the continuous inclusions (see [14], p. 24) which means that T (v, λ) = (ϕ, ψ) ∈ B for all v ∈ B and ∀ λ ∈ [0, 1].
We will continue with the proof of Theorem 2.2.
As a consequence, the uniqueness of solution to problem (4) is valid. Proof. Let g 1 = g 2 = g and w 1 = w 2 = w in the Theorem 2.2. Then (9) shows that the conclusion of the corollary is true.
Remark 1. In order to approximate the unique solution in (1) with homogeneous Neumann boundary conditions ∂ ∂ν ϕ = 0, a scheme of fractional steps type has been introduced and analyzed (convergence and error estimates) in [32].
The results established by Theorem 2.2 highlight the solutions dependence of physical parameters, very useful in the error analysis and numerical simulations.
3. Approximating scheme. Convergence and error estimate. The aim of this Section is to use the fractional steps method in order to approximate the solution of nonlinear boundary value problem (4) (in fact, the solution of problem (1)), whose uniqueness is guaranteed by Corollary 1. Such a method consists in associating to problem (4) for every ε > 0 the following approximating scheme (see also [2], [3], [28], [29], [31]): where z(ε, ϕ ε − (iε, x)) is the solution of Cauchy problem: − stands for the left-hand limit of ϕ ε . We point out that the sequence of approximating problems (38)-(39) supplies a decoupling method for the original problem (4) into a linear parabolic boundary value problem (38) and a nonlinear evolution equation (39). Accordingly, the advantage of this approach consists in simplifying the numerical computation of the process of approximation for the solution of nonlinear problem (1).
The main question is the convergence of the sequence (ϕ ε , ψ ε ) of solutions to the approximate problems (38)-(39) to the unique solution (ϕ, ψ) of problem (4) as ε → 0. We will treat the convergence of this numerical scheme on the basis of compactness (in particular Helly-Foias theorem).
.., M ε − 1, which satisfies (38)-(39) in the following sense: 3.1. Convergence of the approximating scheme. In this section, we will prove the convergence of the iterative scheme (38)-(39) of fractional steps type to the nonlinear parabolic boundary value problem (4). We have . Let (ϕ ε , ψ ε ) be the solution of the approximating scheme (38)-(39). Then for ε → 0, one has is the weak solution to the nonlinear phase transition equation (4).
The following lemmas, which targets the Cauchy problem (39) and which are very useful in the proof of the main result of this Section (Theorem 3.3) were established for the first time in the work [28]. For reader convenience we fully reproduce their proofs.