New approximate method for the Allen – Cahn equation with double-obstacle constraint and stability criteria for numerical simulations

Abstract: In a numerical study, we consider the Allen–Cahn equation with a double-obstacle constraint. The constraint is a multivalued function that is provided by the subdifferential of the indicator function on a closed interval. Therefore, performing a numerical simulation of our problem poses difficulties. We propose a new approximate method for the constraint and demonstrate its validity. Moreover, we present stability criteria for the standard forward Euler method guaranteeing stable numerical experiments when using the approximating equation.


Introduction
For each ε ∈ (0, 1], we consider the following Allen-Cahn equation with double obstacle constraint: ∂u ε ∂ν = 0 on Σ := (0, T ) × Γ, (1.2) where 0 < T < ∞, Ω is a bounded domain in R N (1 ≤ N < +∞) with Lipschitz boundary Γ := ∂Ω, ν is an outward normal vector on Γ and u ε 0 is a given initial value.Also, the double obstacle constraint (1.4) More precisely, ∂I [−1,1] (•) is a set-valued mapping defined by (1.5) The Allen-Cahn equation was proposed to describe the macroscopic motion of phase boundaries.In this physical context, the function u ε = u ε (t, x) in (P) ε :={(1.1),(1.2), (1.3)} is a non-conserved order parameter that characterizes the physical structure.Indeed, let v = v(t, x) be the local ratio of the volume of a pure liquid relative to that of a pure solid at time t and position x ∈ Ω, defined by v(t, x) := lim r↓0 volume of pure liquid in B r (x) at time t |B r (x)| , where B r (x) is the ball in R N with center x and radius r and |B r (x)| denotes its volume.Setting u ε (t, x) := 2v(t, x) − 1 for any (t, x) ∈ Q, we then observe that u ε (t, x) is the non-conserved order parameter that characterizes the physical structure: for the pure liquid region, u ε (t, x) = −1 for the pure solid region, −1 < u ε (t, x) < 1 for the mixed region.
Also, there is a vast literature on the numerical analysis of the Allen-Cahn equation without constraint ∂I [−1,1] (•).For these studies, we refer to [10,11,24,27,28].However, we observe that it is hard to perform a numerical experiment of (P) ε , because the double obstacle constraint ∂I [−1,1] (•) is a multivalued function (cf.(1.5)).Recently, Blank et al. [3] proposed as a numerical method the primal-dual active set algorithm for the local and nonlocal Allen-Cahn variational inequalities with constraint.Also, Farshbaf-Shaker et al. [8] obtained results for the limit of a solution u ε and an element of ∂I [−1,1] (u ε ), called the Lagrange multiplier, to (P) ε as ε → 0.Moreover, they provided the numerical experiment to (P) ε via the Lagrange multiplier for a one-dimensional space for sufficiently small ε ∈ (0, 1] in [9]. The approximate methods are used to obtain a priori estimate of the solutions to (P) ε (cf.[18,21,22,23]).Indeed, the Yosida approximation of ∂I [−1,1] (•) is often used: for δ > 0, the Yosida approximation where [z] + is the positive part of z.Then, by considering approximate problems of (P) ε and letting δ ↓ 0, we can obtain estimates of solutions to (P) ε .Also, using the Yosida approximation (∂I , numerical experiments of the approximate problem of (P) ε were often provided.Recently, in [25], the authors clarified the role of the stability condition in providing stable numerical experiments of the approximate problem of (P) ε via the Yosida approximation in the one-dimensional case.However, we observed that the numerical solution to the approximate equation takes the value outside [−1, 1] (cf.[25,Table 1]).
The plan of this paper is as follows.In Section 2, we discuss the validity of our approximate method defined by (1.7).Indeed, we prove the main result (Theorem 2.1) concerning the solvability and convergence result of (P) ε δ corresponding to item (a) above.In Section 3, we consider (E) ε δ numerically.Then, we prove the main result (Theorem 3.1) corresponding to item (b) above.Also, we provide several numerical experiments to (E) ε δ for small ε ∈ (0, 1] and δ ∈ (0, 1).In the final Section 4, we consider from the view-point of numerical analysis (P) ε δ for a 2D space.Then, we prove the main result (Theorem 4.1) corresponding to item (c) above and provide numerical experiments of (P) ε δ for small ε ∈ (0, 1] and δ ∈ (0, 1).

Notation and basic assumptions
Throughout this paper, we consider the usual real Hilbert space structure denoted by H := L 2 (Ω).The inner product in H is denoted by (•, •) H .We also write V := H 1 (Ω).
In Section 2, we use some techniques of proper (that is, not identically equal to infinity), l.s.c.(lower semi-continuous), convex functions and their subdifferentials, which are useful in the systematic study of variational inequalities.Therefore, let us outline some notation and definitions.For a proper, l.s.c., convex function ψ : H → R ∪ {+∞}, the effective domain D(ψ) is defined by The subdifferential of ψ is a possibly multi-valued operator in H and is defined by z * ∈ ∂ψ(z) if and only if z ∈ D(ψ) and (z * , y − z) H ≤ ψ(y) − ψ(z) for all y ∈ H.
Next, we recall the notion of convergence for convex functions developed by Mosco [19].
Definition 1.1 (cf.[19]).Let ψ, ψ n (n ∈ N) be proper, l.s.c., convex functions on H.Then, we say that ψ n converges to ψ on H in the sense of Mosco [19] as n → ∞, if the following two conditions are satisfied: (M2) for any z ∈ D(ψ), there is a sequence {z n } in H such that

AIMS Mathematics
Volume 1, Issue 3, 288-317 For various properties and related notions of the proper, l.s.c., convex function ψ and its subdifferential ∂ψ, we refer to the monograph by Brézis [4].
Next we present condition (A), which we shall use throughout the paper and assume applies to data.
Proof.Applying the abstract theory of nonlinear evolution equations (cf.[5,15]), we can prove this Proposition 2.1.Indeed, for each ε ∈ (0, 1], we define a functional ϕ ε on H by (2.1) Clearly, ϕ ε is proper, l.s.c. and convex on H with the effective domain Then, the problem (P) ε can be rewritten as an abstract evolution equation of the form: Therefore, applying the Lipschitz perturbation theory of abstract evolution equations (cf.[5,15]), we demonstrate the existence-uniqueness of a solution u ε to (CP) ε , hence (P) ε , on [0, T ] for each ε ∈ (0, 1] in the sense of Definition 2.2.Thus, the proof of Proposition 2.1 is complete.Now, we mention the first main result concerning the solvability and convergence of solutions to (P) ε δ on [0, T ].
and Kδ (•) be convex functions defined by (1.4) and (2.3), respectively.Then, Proof.We first check the condition (M1).Let {δ k } ⊂ (0, 1), {z k } ⊂ R and z ∈ R so that As R is a one-dimensional space, we observe that Now we assume that z > 1.Then, there exists a small positive constant µ such that Then, from (2.5), there exists a number k µ ∈ N satisfying Therefore, we have Hence, we infer from (2.3) and (2.6) that
Thus, the proof of Theorem 2.1 is complete.
By arguments similar to the proof of Theorem 2.1, the following result of (E) ε δ holds.Hence, its detailed proof is omitted.

Stable criteria and numerical experiments for (E) ε δ
Note from (1.5) that ∂I [−1,1] (•) is a multivalued function and therefore very hard to investigate (E) ε numerically.However, we observe from Corollary 2.1 that (E) ε δ is the approximate problem of (E) ε .Hence, in this section, we consider (E) ε δ from the view-point of numerical analysis.We present results concerning numerical experiments of (E) ε δ .There is a vast method on numerical simulations of the ODE problem (e.g., backward Euler scheme, Runge-Kutta method and so on).The authors provided the numerical experiment to (E) ε via the Yosida approximation using the standard forward Euler method in [25].To clarify the advantage of our new approximate method (1.7), we also provide numerical experiments of (E) ε δ using the standard forward Euler method.To this end, we consider the following explicit finite difference scheme to (E) ε δ , denoted by (DE) ε δ : where N t ∈ N is a given positive integer and t := T/N t is the mesh size for time.
Proof.We prove this theorem by arguments similar to those for the proof of [25,Theorem 7].We present the proof only for initial values u ε 0 ∈ (0, 1), because for u ε 0 ∈ (−1, 0) the same arguments apply.
This completes the proof of Theorem 3.1.
From (ii) of Theorem 3.1, we observe that u n oscillates for sufficiently large n and converges to the zero point of f δ (•) for t ∈ δε 2 /(1 − δ), 2δε 2 /(1 − δ) .However, for t = 2δε 2 /(1 − δ), we have the following special case that the solution to (DE) ε δ does not oscillate and coincides with the zero point of f δ (•) after some finite number of iterations. (3.15) Proof.We present only the proof of (3.15) as similar arguments hold for u ε Therefore we infer from (3.2), (3.3), and u 0 = u ε 0 that Similarly, we observe from u 1 ∈ (0, 1 − δ) that Repeating this procedure, we note that from Remark 3.2 the solution to (DE) ε δ is given by (3.15).
Taking into account Theorem 3.1, we present results of numerical experiments of (DE) ε δ .To this end, we take T = 0.002, ε = 0.01, δ = 0.01 and the initial data u ε 0 = 0.1 as numerical data.Then, we observe that: 3.1.The case when t = 0.000001 Setting t = 0.000001, we have: which complies with (i) of Theorem 3.1.Hence, we have the following stable numerical result of (DE) ε δ .Indeed, we observe from Figure 2 and Table 1 in In [25], the authors provided numerical results of the following difference equation: Then, we obtained the following Next we set t = 0.000002 where we have which complies with (ii) of Theorem 3.1.Hence, we observe from Figure 3 and Table 2 that the solution to (DE) ε δ oscillates and converges to the stationary solution 1.  1 − δ Setting t = 2δε 2 /(1 − δ) = 0.0000020202020 • • • , we note Remark 3.1.Indeed, we observe from Figure 4 and Table 3 that the solution to (DE) ε δ oscillates.
which implies that the criteria condition (4.4) holds.Thus, we obtain a stable numerical experiment of (DP) ε δ yielding Figure 9: Therefore, the criteria (4.4) does not hold and hence yields an unstable numerical experiment of (DP) ε δ .Indeed, we obtain Figure 10: which implies that the inequalities (4.4) hold.Therefore, we obtain a stable numerical experiment of (DP) ε δ that yields Figure 11: δ are produced if we choose suitable values for the constants ε, δ, and mesh sizes for time t and space x and y.Therefore, if we perform a numerical experiment of (P) ε for sufficiently small ε, we found it imperative to consider the original problem (P) ε using the primal-dual active set method of [3], the Lagrange multiplier method of [9] and so on.

Table 1 .
Numerical results of (DE) ε δ and (YDE) ε δ for t = 0.000001.number of iterations i the value of u i to (DE) ε

Table 2 .
Numerical result of (DE) ε δ for t = 0.000002.number of iterations i the value of u i