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 O(M log M ) and O(M), 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 K,η on the width of the interfaces.
Highlights
As a typical phase-field model, the classical Allen-Cahn equation [1]∂tu − KΔu + ψ (u) = 0, (1)was originally derived through the minimization of the Ginzburg-Landau free energy functional ∫ Ω ( |∇u|2 + ψ (u)) dx (2)to describe the motion of antiphase boundaries in crystalline solids with the double-well potential ψ(u) = (1/4η2)(u2 − 1)2
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 AllenCahn 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
We find that if the linear finite element space is employed, the matrix B is a Toeplitz matrix, which makes it possible to reduce the computation cost and storage to O(M log M) by a delicate combination of the conjugate gradient (CG) algorithm, the fast Fourier transform (FFT) and its Toeplitz structure of the matrix B
Summary
Was originally derived through the minimization of the Ginzburg-Landau free energy functional. 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 AllenCahn 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. Numerical experiments are conducted to test the efficiency of the proposed efficient finite element algorithm
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