Abstract
We embed the fractional Allen-Cahn equation into a Galerkin variational framework and thus develop its corresponding finite element procedure and then prove rigorously its mathematical and physical properties for the finite element solution. Combining the merits of the conjugate gradient (CG) algorithm and the Toeplitz structure of the coefficient matrix, we design a fast CG for the linearized finite element scheme to reduce the computation cost and the storage to and respectively. Numerical experiments confirm that the proposed fast CG algorithm recognizes accurately the mass and energy dissipation, the phase separation through a very clear coarse graining process, and the influences of different indices r of fractional Laplacian and different coefficients on the width of the interfaces.
1. Introduction
As a typical phase-field model, the classical Allen-Cahn equation [1]was originally derived through the minimization of the Ginzburg-Landau free energy functional to describe the motion of antiphase boundaries in crystalline solids with the double-well potential . Since then, the Allen-Cahn equation has been widely applied to many complicated moving interface problems, for example, vesicle membranes, nucleation of solids, and mixture of two incompressible fluids, etc. (cf. [2–8]), and many and many research results on its theories, applications, and numerics of the Allen-Cahn equation have been achieved; see the reviews [9–15] and the references cited therein.
To recognize the influences of the long-range interactions between particles in those complicated moving interface problems, it could reasonably make physical significance if the Laplacian operator in (1) is replaced by its fractional version of Riesz-type potential to form the fractional Allen-Cahn equation. In this line, [16] proposed a kind of fractional Allen-Cahn model and discussed the solvability in some fractional Sobolev spaces, and [17] developed a fractional extension of the Allen-Cahn phase-field model with its fractional Laplacian defined by Riemann-Liouville fractional derivative that describes the mixture of two incompressible fluids. In [17], the authors also proposed a Petrov-Galerkin spectral method for spacial discretization combined with a stabilized ADI scheme for temporal discretization in the absence of rigorous numerical analysis for the solvability, stability, convergence, and conservation properties of the numerical scheme. Some other numerical methods such as finite difference [18], finite volume [19], and collocation method [20] were established. As far as we know, few research works have been done on the efficient finite element method and its rigorous numerical analysis for the fractional Laplace operator defined as Riesz-type potentials on the whole space
In this article, we consider the following fractional Allen-Cahn equation [16]:where , is the diffusion coefficient and the fractional Laplace operator is defined as Riesz-type potentials on the whole space in Section 2. The unknown can be viewed as an indicator of the concentration or volume fraction of one fluid at the location in the immiscible mixture with the second fluid.
The main objectives of this article are to (1) embed the fractional Allen-Cahn equation (1) into a Galerkin variational framework and thus develop its corresponding finite element procedure; (2) prove rigorously the solvability, the optimal convergence rates, and the physical properties, for examples, the mass decay, the energy dissipation, and the new energy equality, for the finite element solution; (3) combine the Toeplitz structure of the coefficient matrix and the merits of the classic CG algorithm [21] to design a fast CG (FCG) for the linearized finite element scheme, which reduces the computation cost and the storage to and , respectively; and (4) conduct numerical experiments to verify the efficiency of the FCG, which show that the FCG possesses the ideal convergence rates as Newtons algorithm does in space and time, preserves the mass, energy dissipation, and energy equality law, and recognizes accurately the phase separation by a very clear coarse graining process. The numerical experiments also test the tunable sharpness, that is, the influences of different fractional indices and different coefficients .
The rest of this article is outlined as follows. Section 2 is preliminaries. Section 3 is for the solvability and stability of the solution of discrete system. We demonstrate that the discrete solution preserves mass and energy dissipation and satisfies new energy equality in Section 4. Sections 5 and 6 are devoted to convergence analysis and efficient FCG algorithm, respectively. In the last section, numerical experiments are conducted to test the efficiency of the proposed efficient finite element algorithm.
2. Preliminaries
We first briefly revisit the definitions and some properties of fractional Laplace operator.
Definition 1 (see [22, 23]). For and , where is the Schwartz class of rapidly decaying functions at infinity, the fractional Laplace operator is reformulated bywhere is a constant given by
Definition 2 (see [22, 24, 25]). For , the fractional Sobolev spaces are defined by and equipped with the norm and with equivalent seminorm The energy space and the energy are defined, respectively, by
Definition 3 (see [22]). For , the operator is defined by Here stands for its dual of .
It is easily seen that is a symmetric positive definite operator.
Definition 4 (see [22]). Let If satisfiesthen the is called a weak solution to problem (3).
3. Finite Element Procedure
In this section, we construct finite element scheme for (3) based on the weak formulation (12) and prove the solvability and stability of solution of discrete system.
Multiplying (12) by any , integrating over and combining the homogeneous boundary condition, we obtain the variational formulation of (3) as to find such that
Taking and as integers, we divide uniformly by intervals for with , , , and partition the time interval by the nodes for with the time step .
Upon the space partition, we define the finite element space as here being the set of polynomials of degree not bigger than over the interval .
Applying the backward Euler scheme to discrete the time derivative , we define the fully discrete finite element procedure of (13) as to find such that, for ,where is the elliptic projection of the initial value to the finite element space.
The dimension of is ; see the review [24]. Assume are the basis functions; then, we can express the numerical solution byand, thus, the fully discrete finite element scheme (15) is transformed equivalently to the following algebraic equation:where is the unknown vector, and the matrices are It is easily verified that , , and are symmetric and positive matrices.
By using the contraction mapping principle, we prove the existence and uniqueness of the fully discrete finite element scheme (15).
Theorem 5. There exists a unique solution to (17) for sufficiently small .
Proof. Selecting the time step sufficiently small such that and noticing that and the matrices and are positive definite, we know that the matrix is positive definite and thus invertible for sufficiently small . Therefore, we solve from (17) Define the mapping : by The mapping is well defined for given due to the positiveness of the matrix
We shall use a corollary of the well-known contraction mapping principle [26] to prove the mapping having a unique fixed point in a bounded domain of
For this purpose, we let and for Then, Noticing that the matrix is Lipschitz continuously with respect to , we obtain Select to be small enough such that and we have which shows that the mapping is a contractive mapping. This, together with an application of the well-known contraction mapping principle, completes the proof.
Theorem 6. Assume that is small enough such that the conditions of Theorem 5 hold. Then, the fully discrete finite element scheme (15) is stable in the following sense, for ,
Proof. Take in (15) to obtain Noting the fact that and we have Adding all the terms from to , we get Namely, Applying the discrete Gronwall inequality, we have This completes the proof.
4. Properties Preserved by the Finite Element Solution
In this section, we demonstrate that the finite element solution preserves the energy dissipation law and satisfies a redefined energy equality.
Theorem 7. The finite element solution of (15) preserves the energy dissipation law in the following sense, for ,
Proof. Taking in (15), we haveDue to the fact that we have and By applying the inequality , we get Then, combining the above inequalities with (35), we deriveTherefore, we have It follows that for . This completes the proof.
We prove the following new energy equality preserved by the finite element solution .
Theorem 8. The finite element solution of (15) satisfies the following energy equality for ,where the left-hand side of the energy equality (42) is redefined as the new energy.
Proof. Choosing and multiplying to both sides in (15), we getBy a simple calculation, we have and Substituting the above equalities into (43) and combining (10), we have Add all terms from to to obtain This completes the proof.
5. Convergence Analysis
In this section, we shall conduct convergence analysis for the fully discrete finite element scheme (15).
Define the elliptic projection of the exact solution as The following estimate is valid for and [27],
Let and in (13), and subtract it from (15) to obtain the error equationDenote then
Theorem 9. Assume that . Then, there exists integers and constants and independent of the step parameters and such that Here
Proof. Take in (50) to obtainWe first estimate the left-hand side of (55).
Applying the inequality and the Young inequality, we have By using the monotonicity of function , we have And then, we estimate the right-hand side of (55). Combining the inequality and the Young inequality, we have Similarly, we get The third term on the right-hand side of (55) is estimated by and Then, taking these inequalities into (55), we obtain Noticing and adding all the terms from to , we have The discrete Gronwall inequality is applied to (63), when is small enough to satisfy , If we take ,,, thenThen, using the elliptic projection estimate (49) and the triangle inequality, we obtainThis concludes the proof.
Remark 10. Since our proof for the stability, energy dissipation, new energy-equality and error deduction of the discrete scheme (15) is independent of the dimensional argument, the analysis and conclusions in this article can be extended to multidimensional models, maybe with minor modification.
6. Fast-Conjugate-Gradient Algorithm (FCG)
The fully discrete finite element scheme (15) is nonlinear and can be solved by Newton’s iteration algorithm. Here we linearize the nonlinear terms by replacing by and solve this linearized version by the conjugated gradient algorithm (CG). We find that if the CG algorithm is directly used, the computation cost and storage will reach up to since the coefficient matrix is nonsparse due to the nonlocality of the fractional Laplace operator. We also find that if the linear finite element space is employed, the matrix is a Toeplitz matrix (see Section 7.1), which makes it possible to reduce the computation cost and storage to by a delicate combination of the CG algorithm, the fast Fourier transform (FFT) and its Toeplitz structure of the matrix
In this section, we depict the general ideas for the combination of CG, FFT, and the Toeplitz matrix and design a fast CG algorithm (FCG). For a good review of a fast algorithm generating from FFT and the Toeplitz matrix, we refer to [28].
Noticing what causes the computation cost up to is a direct use of the matrix-vector multiplication in the classical CG algorithm; as a remedy, we can obtain the matrix-vector multiplication for a vector and Toeplitz matrix through the following two steps [29, 30]:
(1) assembling the Toeplitz matrix into a cyclic matrix ;
(2) extracting the Toeplitz matrix-vector multiplication for a vector via
In (67), is determined simply by the matrix and the computational cost and the storage of the matrix-vector multiplication are and [30]. This will reduce the computation cost and the storage from to and compared to the direct use of matrix-vector multiplication.
Upon this analysis, we write down the improved algorithm, called the fast CG (FCG), sentence by sentence in Algorithm 1, and present the corresponding conclusion by Theorem 11.
| function FCG() ; | |
| ; ; | |
| ; ; | |
| ; ; | |
| ; ; | |
| ; | |
| ; if then | |
| if then ; | |
| ; else | |
| else break; | |
| ; end if | |
| ; end function | |
| end if |
Theorem 11. The computational cost and storage of the fast conjugated gradient algorithm (FCG) are reduced from and to and compared with the traditional CG algorithm.
7. Numerical Experiments
In this section, we linearize the nonlinear term in the fully discrete finite element scheme (15) by taking to replace ; we carry out two numerical experiments to verify the result convergence analysis and the physical property of the solution of discrete system.
7.1. The Linear Finite Element System
For simplicity, we take the linear finite element system to compute algebraic equation (17).
For , the inner nodal basis function is at , with the following structure: For , the matrix A is We replace the nonlinear term with , After calculation, the entries of the matrix areFor , the entries of the matrix are respectivelyAfter calculating the above integrals (72), we obtain that the entries of matrix are solved for where For ,
7.2. Tests on the Efficiency of the Finite Element Procedure and the FCG
Example 12. Assuming , the analytic solution is where , is selected as close to as possible. The source term should be expressed as follows:We apply Example 12 to verify the error convergence rate of the linear finite element system and to test the efficiency of the FCG for .
The numerical solution is calculated by the FCG for . The numerical solution and the exact solution are identical at in Figure 1, which show that the FCG is accurate.
Table 1 show the spatial convergence rates with respect to for respectively. By comparing the convergence rates of FCG to the convergence rate of Newton, Gauss, and CG, respectively, we conclude that the FCG algorithm is efficient. The convergence rates of are at least 1.6 and 1.2 for , respectively, which are almost equal to the theoretical error rate .
Table 2 shows the spatial convergence rates with respect to for , respectively. The convergence rates of are at least 1.9 and 1.5 for , respectively. We observe that the spatial convergence rates decrease as the spatial fractional order increases.
Remark 13. Recalling error estimate for second order elliptic equation, one can obtain the estimate of one order higher than that for . However, in the case of the fractional order elliptic equation, we only obtain the convergence rate of dependent on the index . From the numerical Table 2, the convergence order of conforms to the following relation.
Figure 2 shows that the convergence rate of dependent on the index , as the index increases, the rates of spatial convergence decreases and then increases.
Table 3 shows the temporal convergence rates as , respectively. The temporal convergence rates are almost equal to 1 for at , which are consistent with the theoretical expectation for the backward Euler scheme.
Table 4 shows the efficiency of the FCG algorithm. It is easily obtained that the CPU time consumed by the FCG is much less than that of the Newton, Gauss, and CG as decreases. It can be seen that the CPU time of FCG is minutes while the Newton’s hours and the CG’s hours for , . These show that the FCG is an efficiency algorithm.
7.3. Tests on Physical Properties of Finite Element Solution
Example 14. Letting , , the initial value isThis example is used to test the physical properties such as the energy laws and the new energy equality, the mass dissipation, the phase separation, and the tunable sharpness in Figures 3–8.
We present the energy dissipation in Figure 3 and the new energy equality in Figure 4 for , , and . In Figure 3, the energy dissipates to zero as time increases. In Figure 4, the energy jumps in scale at the beginning until and keeps the same at the end; this demonstrates that the finite element solution to (15) almost satisfies new equality equation.
We plot the mass changes for , , and in Figure 5. The mass value dissipates as time increases; this property is coinciding with the classical Allen-Cahn equation, and this also explains the applicability of fractional Laplacian for Allen-Cahn equation in some way.
We present the physical phenomenon of a coarse graining process as the time increases for , , and in Figure 6. The two phases are connected in a smooth steep line; this indicates that the phase separation is complete.
Figure 7 shows that the interface gets sharper and sharper as decreases. Figure 8 shows that the interface gets sharper and sharper as decreases. The above two groups of experimental results show that the tunable sharpness of the interface is related to different parameters and .
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported in part by the National Natural Science Foundation of China under Grant nos. 11471196, 10971254, and 11471194.