Efficient Implementation and Numerical Analysis of Finite Element Method for Fractional Allen-Cahn Equation

Guozhen Wang and Huanzhen Chen

Mathematical Problems in Engineering, 2019, vol. 2019, 1-14

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.

Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlmpe:7969371

DOI: 10.1155/2019/7969371

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