Abstract
We consider a one-dimensional Allen-Cahn equation with constraint from the viewpoint of numerical analysis. The constraint is provided by the subdifferential of the indicator function on the closed interval, which is the multivalued function. Therefore, it is very difficult to perform a numerical experiment of our equation. In this paper we approximate the constraint by the Yosida approximation. Then, we study the approximating system of the original model numerically. In particular, we give the criteria for the standard forward Euler method to give the stable numerical experiments of the approximating equation. Moreover, we provide the numerical experiments of the approximating equation.
1. Introduction
In this paper, for each we consider the following Allen-Cahn equation with constraint from the viewpoint of numerical analysis: where and is given initial data. Also, the constraint is the subdifferential of the indicator function on the closed interval defined by More precisely, is a set-valued mapping defined by
The Allen-Cahn equation was proposed to describe the macroscopic motion of phase boundaries. In the physical context, the function in is the nonconserved order parameter that characterizes the physical structure. For instance, let be the local ratio of the volume of pure liquid relative to that of pure solid at time and position , defined by where is the ball in with center and radius and denotes its volume. Put for any . Then, we observe that is the nonconserved order parameter that characterizes the physical structure:
There is a vast amount of literature on the Allen-Cahn equation with or without constraint . For these studies, we refer to [1–16]. In particular, Chen and Elliott [5] considered the singular limit of as in the general bounded domain .
Also, there is a vast amount of literature on the numerical analysis of the Allen-Cahn equation without constraint . For these studies, we refer to [17–21].
Note that the constraint is the multivalued function. Therefore, it is very difficult to apply the numerical methods developed by [17–21] to . Hence, it is difficult to perform a numerical experiment of . Recently, Blank et al. [2] proposed as a numerical method a primal-dual active set algorithm for the local and nonlocal Allen-Cahn variational inequalities with constraint. Also, Farshbaf-Shaker et al. [8] gave the results of the limit of a solution and an element of , called the Lagrange multiplier, to as . Moreover, they [9] gave the numerical experiment to via the Lagrange multiplier in one dimension of space for sufficient small . Furthermore, they [9] considered the approximating method, called the Yosida approximation. More precisely, for , the Yosida approximation of is defined by where is the positive part of . Then, for each , they [9] considered the following approximation problem of : In [9, Remark ], Figure 1 shows the unstable numerical result to was given by the standard explicit finite difference scheme to .
From Figure 1, we observe that we have to choose the suitable constants and and the mesh size of time and space in order to get the stable numerical results of . Therefore, in this paper, for each and , we give the criteria for the standard explicit finite difference scheme to provide the stable numerical experiments of . To this end, we first consider the following ODE problem, denoted by : Then, we give the criteria to get the stable numerical experiments of . Also, we give some numerical experiments of . Moreover, we show the criteria to get the stable numerical experiments of PDE problem . Therefore, the main novelties are the following: (a)We give the criteria to get the stable numerical experiments of the ODE problem . Also, we provide the numerical experiments to for sufficient small and .(b)We give the criteria to get the stable numerical experiments of the PDE problem . Also, we provide the numerical experiments to for sufficient small and .
The plan of this paper is as follows. In Section 2, we mention the solvability and convergence result of . In Section 3, we consider numerically. Then, we prove the main result (Theorem 7) corresponding to item (a) listed above. Also, we provide the numerical experiments to for sufficient small and . In Section 4, we mention the solvability and convergence result of . In the final Section 5, we consider from the viewpoint of numerical analysis. Then, we prove the main result (Theorem 16) corresponding to item (b) listed above. Also, we provide the numerical experiments to for sufficient small and .
Notations and Basic Assumptions. Throughout this paper, we put with the usual real Hilbert space structure. The inner product and norm in are denoted by and by , respectively. We also put with the usual norm for
In Sections 2 and 4, we use some techniques of proper (i.e., not identically equal to infinity), l.s.c. (lower semicontinuous), and convex functions and their subdifferentials, which are useful in the systematic study of variational inequalities. So, let us outline some notations and definitions. Let be the real Hilbert space with the inner product . For a proper, l.s.c., and convex function , the effective domain is defined by The subdifferential of is a possibly multivalued operator in and is defined by if and only if For various properties and related notions of the proper, l.s.c., and convex function and its subdifferential , we refer to the monograph by Brézis [22].
2. Solvability and Convergence Results of
We begin by giving the rigorous definition of solutions to our problem ( and ).
Definition 1. Let , , and . Then, a function is called a solution to on , if the following conditions are satisfied:(i)(ii)The following equation holds: (iii) in .
Now, we show the solvability result of on .
Proposition 2. Let , , and with . Then, there exists a unique solution to on in the sense of Definition 1.
Proof. We can prove the uniqueness of solutions to on by the quite standard arguments: monotonicity and Gronwall’s inequality.
Also, we can show the existence of solutions to on applying the abstract theory of evolution equations governed by subdifferentials. Indeed, we define a function on by Clearly, is proper, l.s.c., and convex on with in , where is the Yosida approximation of defined by (8).
Note that the problem can be rewritten as in an abstract framework of the form Therefore, applying the Lipschitz perturbation theory of abstract evolution equations (cf. [23–25]), we can show the existence of a solution to , hence, , on for each and in the sense of Definition 1. Thus, the proof of Proposition 2 has been completed.
Next, we show the convergence result of as . To this end, we recall a notion of convergence for convex functions, developed by Mosco [26].
Definition 3 (cf. [26]). Let and () be proper, l.s.c., and convex functions on a Hilbert space . Then, one says that converges to on in the sense of Mosco [26] as , if the following two conditions are satisfied: (i)for any subsequence , if weakly in as , then (ii)for any , there is a sequence in such that
It is well known that the following lemma holds. Therefore, we omit the detailed proof.
Lemma 4 (cf. [27, Section ], [22, Chapter ], and [28, Section ]). Consider
By Lemma 4 and the general convergence theory of evolution equations, we get the following result. We omit the detailed proof.
Proposition 5 (cf. [27, Section ], [28, Section ]). Let , , and with . Also, let be the unique solution to on . Then, there exists a unique function such that and is the unique solution of the following problem on :
3. Stable Criteria and Numerical Experiments for
In this section we consider from the viewpoint of numerical analysis.
Remark 6. Note from Proposition 5 that is the approximating problem of . Also note from (5) that the constraint is the multivalued function. Therefore, it is very difficult to study numerically.
In order to perform the numerical experiments of via the standard forward Euler method, we consider the following explicit finite difference scheme to , denoted by :where is the mesh size of time and is the integer part of number .
We observe that is the approximating solution of at the time . Also, we observe that the explicit finite difference scheme converges to as since is the standard time discretization scheme for .
In Figure 2, we give the unstable numerical experiment of in the case when , , , the initial data , and the mesh size of time .
From Figure 2, we observe that we have to choose the suitable constants and and the mesh size of time in order to get the stable numerical results of .
Now, let us mention the first main result in this paper, which is concerned with the criteria to give the stable numerical experiments of .
Theorem 7. Let , , and . Assume (resp., ) and . Let be the solution to . Then, one has the following: (i)If , converges to (resp., ) monotonically as .(ii)If , oscillates and converges to (resp., ) as .
Proof. We give the proof of Theorem 7 in the case of the initial value .
For simplicity, we set Then, we observe that and , , are the zero points of . Also, we observe that the difference equation is reformulated in the following form: Note from (23) and (24) that if we have which implies that is increasing with respect to until .
Now, we prove (i). To this end, we assume that . At first, by the mathematical induction, we show Clearly (26) holds for because of .
Now, we assume that (26) holds for all . Suppose . Then, we infer from (25) that Therefore, by (25) and the inequality as above, we observe that Next, if , we observe from (23) and (24) that which implies that From (28) and (30) we infer that (26) holds for . Therefore, we conclude from the mathematical induction that (26) holds.
Also, by (23) and (26) we observe that for all . Therefore, we observe from (24) that Therefore, we infer from (26) and (31) that is a bounded and increasing sequence with respect to . Thus, there exists a point such that Taking the limit in (24), we observe from the continuity of that , which is the zero point of . Hence, the proof of (i) has been completed.
Next, we show (ii). To this end, we assume that . Then, we can find the minimal number so that Indeed, if for all , we observe from (25) that Taking into account (34), , and we can find the minimal number so that Also, we observe from (25) that and thus, (33) holds.
To show (ii), we put Then, we observe from (23) and (24) that From (39) it follows that Therefore, we observe from (40) and that the zero point of is in the interval between and .
Also, by (39) we observe that which implies that Therefore, by (33) and (40) and by repeating the procedure as above, we observe that and oscillates around the zero point for all .
Also, we observe from (39) and (43) that Therefore, by , (43), and (44), there is a subsequence of such that oscillates and converges to as . Hence, taking into account the uniqueness of the limit point, the proof of (ii) has been completed.
Remark 8. Assume and put for some . Then, we observe that Therefore, we infer from the proof of Theorem 7 (cf. (34), (40), and (44)) that the solution to oscillates as , in general.
Remark 9. By (24) we easily observe that
From (ii) of Theorem 7, we observe that oscillates and converges to the zero point of in the case when . However, in the case when , we have the following special case that the solution to does not oscillate and coincides with the zero point of after some finite number of iterations.
Corollary 10. Let , , , and . Assume . Then, the solution to is given by
Proof. Note that . Therefore, by (23) and (24) we observe that Similarly, we have Repeating this procedure, we observe that the solution to is given by (47).
Taking into account Theorem 7, we provide the numerical experiments of as follows. To this end, we use the following numerical data.
Numerical Data of The numerical data are , , , and the initial data . Then, we observe that
3.1. The Case When
Now we consider the case when . In this case, we have which implies that (i) of Theorem 7 holds. Thus, we have the stable numerical result of . Indeed, we observe from Figure 3 and Table 1 that the solution to converges to the stationary solution .
3.2. The Case When
Now we consider the case when . In this case, we have which implies that (ii) of Theorem 7 holds. Thus, we observe from Figure 4 and Table 2 that the solution to oscillates and converges to the stationary solution .
3.3. The Case When
Now we consider the case when . In this case, we observe Remark 8. Indeed, we observe from Figure 5 and Table 3 that the solution to oscillates.
3.4. The Case When
Now we consider the case when . In this case, we have Therefore, we observe Remark 8. Indeed, we observe from Figure 6 that the solution to oscillates.
3.5. The Case When
Now we consider the case when . In this case, we observe Remark 8. Indeed, we observe from Figure 7 that the solution to oscillates between three zero points of .
3.6. Numerical Result of Corollary 10
In this subsection, we consider Corollary 10 numerically. To this end, we use the following initial data: Then, we observe from Table 4 and Figure 8 that Corollary 10 holds. Namely, we observe that (47) holds with .
3.7. Conclusion of ODE Problem
From Theorem 7 and numerical experiments as above, we conclude that(i)the mesh size of time must be smaller than in order to get the stable numerical experiments of ,(ii)we have the stable numerical experiments of with the initial data , even if the mesh size of time is equal to .
4. Solvability and Convergence Results for
We begin by giving the rigorous definition of solutions to our PDE problem ( and ).
Definition 11. Let , , and . Then, a function is called a solution to on , if the following conditions are satisfied: (i).(ii)The following variational identity holds: (iii) in . Now, we mention the solvability result of on .
Proposition 12. Let and . Assume the following condition: Then, for each , there exists a unique solution to on in the sense of Definition 11.
Proof. By the same argument as in Proposition 2, we can show the existence-uniqueness of a solution to on for each and . Indeed, we can prove the uniqueness of solutions to on by the quite standard arguments: monotonicity and Gronwall’s inequality.
Also, we can show the existence of solutions to on applying the abstract theory of evolution equations governed by subdifferentials. Indeed, we define a functional on by where is the function defined in (14). Clearly, is proper, l.s.c., and convex on with the effective domain .
Note that the problem can be rewritten as in an abstract framework of the form Therefore, applying the Lipschitz perturbation theory of abstract evolution equations (cf. [23–25]), we can show the existence of a solution to , hence, , on for each and in the sense of Definition 11. Thus, the proof of Proposition 12 has been completed.
Next, we recall the convergence result of as . Taking into account Lemma 4 (cf. (18)), we observe that the following lemma holds.
Lemma 13 (cf. [27, Section ], [22, Chapter ], [28, Section ]). Let , and define a functional on by Then,
By Lemma 13 and the general convergence theory of evolution equations, we get the following result.
Proposition 14 (cf. [27, Section ], [28, Section ]). Let , , and . Also, let be the unique solution to on . Then, converges to the unique function to on in the sense that
Proof. Note that the problem can be rewritten as in an abstract framework of the form Therefore, by Lemma 13 and the abstract convergence theory of evolution equations (cf. [27, 28]), we observe that the solution to converges to the unique solution to on as in the sense of (60). Note that (resp., ) is the unique solution to (resp., ) on (cf. Proposition 12). Thus, we conclude that Proposition 14 holds.
5. Stable Criteria and Numerical Experiments for
In this section we consider from the viewpoint of numerical analysis.
Remark 15. Note from Proposition 14 that is the approximating problem of . Also note from (5) that the constraint is the multivalued function. Therefore, it is very difficult to study numerically.
In order to perform the numerical experiments of , we consider the following explicit finite difference scheme to , denoted by : where is the mesh size of time, is the mesh size of space, is the integer part of number , is the integer part of number , and .
We observe that is the approximating solution of at the time and the position . Also, we observe that the explicit finite difference scheme converges to as and .
From Figure 1, we observe that we have to choose the suitable constants and and the mesh size of time and the mesh size of space in order to get the stable numerical results of . Now, let us mention our second main result concerning the stability of .
Theorem 16. Let , , , , , and , where is the set of initial values defined in Proposition 12 (cf. ). Let be the integer part of number , and let be the solution to . Also, let and assume that Then, one has the following: (i)The solution to is bounded in the following sense: (ii) does not oscillate with respect to .
Proof. We first show (i), that is, (64), by the mathematical induction.
Clearly (64) holds for because of .
Now, we assume that We observe that the explicit finite difference problem can be reformulated as in the following form: where we put and is the function defined by (23).
We observe from (63), (65), and (66) that From (23), (63), and (65) we infer that the function is nonnegative and continuous. Indeed, it follows from (23) that the function attains a minimum value at . Therefore, we observe from (23) and (63) thatAlso, for any , we observe from (23) that Here we note from (63) that Therefore, we infer from (69) that the function is nonincreasing and attains a minimum value at . Hence, we have Thus, we observe from (68) and (71) that which implies from (65) and (67) that Similarly, we observe from (63), (65), and (66) that By similar arguments as above, we observe that the function is nonnegative and continuous. In fact, it follows from (23) that the function attains a minimum value at . Therefore, we observe from (23) and (63) that Also, for any , we observe from (23) that Here we note from (63) that Therefore, we infer from (76) that the function is nondecreasing and attains a minimum value at . Hence, we have Thus, we observe from (75) and (78) that which implies from (65) and (74) that Taking into account Neumann boundary condition, namely, by and , we observe from (73) and (80) that which implies that (65) holds for . Therefore, we conclude from the mathematical induction that (64) holds. Hence, the proof of (i) of Theorem 16 has been completed.
Next, we show (ii) by the standard arguments. Namely, we reformulate as in the following form: Here by taking into account Neumann boundary condition and initial condition, namely, by and for all , we observe that (82) is reformulated as in the following form: where we put Note from (23) that Defining we observe from (85) and (86) that Using the matrix as above, (82) can be rewritten as in the following form: where we define By the general theory, we observe that the eigenvalue of matrix is given by which implies that (resp., ) is the maximum (resp., minimum) eigenvalue of .
Now let be the set of all eigenvalues of matrix such that Also, let be the set of all eigenvalues of such that Then, by the abstract perturbation theory of matrix, we observe that Since is the symmetric matrix, it follows from (63), (86), and (90) that Thus, we conclude from (93) and the above inequality that Note that (95) holds in any case: and .
Next, we now assume . Then, we observe from (63) and (86) that Therefore, all components of are positive. Hence, the sum of all components in the th row of is the following:Therefore, we observe from (97) that is increasing with respect to in the case when .
However, if for some , it follows from (63) and (86) that Therefore, the sum of all components in the th row of is the following: Although is increasing with respect to in the case when (cf. (97)), we conclude from (95) and (99) that (ii) of Theorem 16 holds. Thus, the proof of Theorem 16 has been completed.
Remark 17. By (63) we get the suitable mesh size of space . In fact, for each , we take the constant , , such that Then, we have Thus, we have the following condition of :
Remark 18. We can take in (63) for the explicit finite difference scheme to the following usual heat equation with Neumann boundary condition:
Taking into account Theorem 16, we provide the numerical experiments of as follows. To this end, we use the following numerical data.
Numerical Data of The numerical data is , , , . Also, we consider the initial data defined by
5.1. The Case When and
Now, we consider the case when and . In this case, we observe that Therefore, we have which implies that criteria condition (63) holds. Thus, we can provide the stable numerical experiment of . Indeed, we provide Figure 9, which implies the stable numerical experiment of .
5.2. The Case When and
Now, we consider the case when and . In this case, we observe that Therefore, we have which implies that criteria condition (63) does not hold. Therefore, we provide the unstable numerical experiment of . Indeed, we have Figure 10.
5.3. The Case When and
Now, we consider the case when and . In this case, we have which implies that criteria condition (63) holds. Therefore, we provide the stable numerical experiment of . Indeed, we have Figure 11, which implies the stable numerical experiment of .
Remark 19. We observe from Theorem 16 that, in order to get the stable numerical results of , we have to choose the suitable constants and and the mesh size of time and space . Therefore, if we perform a numerical experiment of for sufficient small , we had better consider the original problem by using a primal-dual active set method in [2], a Lagrange multiplier method in [9], and so on.
5.4. Conclusion of PDE Problem
By Theorem 16 and the numerical experiments as above, we conclude that the mesh size of time and space must be satisfied: in order to get the stable numerical experiments of . Also, by Theorems 7 and 16, we conclude that the value is very important to perform a numerical experiment of and .
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work was supported by Grant-in-Aid for Scientific Research (C) no. 26400179, JSPS.