On Diagonal Dominance of FEM Stiffness Matrix of Fractional Laplacian and Maximum Principle Preserving Schemes for the Fractional Allen–Cahn Equation

In this paper, we study diagonal dominance of the stiffness matrix resulted from the piecewise linear finite element discretisation of the integral fractional Laplacian under global homogeneous Dirichlet boundary condition in one spatial dimension. We first derive the exact form of this matrix in the frequency space which is extendable to multi-dimensional rectangular elements. Then we give the complete answer when the stiffness matrix can be strictly diagonally dominant. As one application, we apply this notion to the construction of maximum principle preserving schemes for the fractional-in-space Allen-Cahn equation, and provide ample numerical results to verify our findings.

[1]  Qiang Du,et al.  Time-Fractional Allen–Cahn Equations: Analysis and Numerical Methods , 2019, Journal of Scientific Computing.

[2]  Han Zhou,et al.  A class of second order difference approximations for solving space fractional diffusion equations , 2012, Math. Comput..

[3]  Andreas Prohl,et al.  Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows , 2003, Numerische Mathematik.

[4]  Shuonan Wu,et al.  On the stability and accuracy of partially and fully implicit schemes for phase field modeling , 2016, Computer Methods in Applied Mechanics and Engineering.

[5]  Tao Tang.  Revisit of Semi-Implicit Schemes for Phase-Field Equations , 2020, ArXiv.

[6]  Jie Shen,et al.  Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains , 2019, SIAM J. Numer. Anal..

[7]  I. Babuska,et al.  The finite element method and its reliability , 2001 .

[8]  J. Rencis,et al.  The importance of diagonal dominance in the iterative solution of equations generated from the boundary element method , 1993 .

[9]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[10]  K. Burrage,et al.  Fourier spectral methods for fractional-in-space reaction-diffusion equations , 2014 .

[11]  G. Karniadakis,et al.  A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations , 2016 .

[12]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[13]  Alan George,et al.  Gaussian elimination is stable for the inverse of a diagonally dominant matrix , 2003, Math. Comput..

[14]  Yanzhi Zhang,et al.  Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications , 2019 .

[15]  Q. Du,et al.  The phase field method for geometric moving interfaces and their numerical approximations , 2019, Geometric Partial Differential Equations - Part I.

[16]  Tao Tang,et al.  Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations , 2016, J. Sci. Comput..

[17]  Qiang Ye,et al.  Accurate computation of the smallest eigenvalue of a diagonally dominant M-matrix , 2002, Math. Comput..

[18]  Mark Ainsworth,et al.  Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains , 2018 .

[19]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[20]  Tao Zhou,et al.  On Energy Dissipation Theory and Numerical Stability for Time-Fractional Phase-Field Equations , 2018, SIAM J. Sci. Comput..

[21]  Sören Bartels,et al.  Numerical Approximation of Partial Differential Equations , 2016 .

[22]  Gabriel Acosta,et al.  A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations , 2015, SIAM J. Numer. Anal..

[23]  Yukun Li,et al.  Numerical Methods for Deterministic and Stochastic Phase Field Models of Phase Transition and Related Geometric Flows , 2015 .

[24]  Hong Wang,et al.  On power law scaling dynamics for time-fractional phase field models during coarsening , 2018, Commun. Nonlinear Sci. Numer. Simul..

[25]  Gabriel Acosta,et al.  A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian , 2016, Comput. Math. Appl..

[26]  Farid O. Farid,et al.  Notes on matrices with diagonally dominant properties , 2011 .

[27]  Jiang Yang,et al.  Implicit-Explicit Scheme for the Allen-Cahn Equation Preserves the Maximum Principle , 2016 .

[28]  Feng Wang,et al.  Finite element simulation and efficient algorithm for fractional Cahn-Hilliard equation , 2019, J. Comput. Appl. Math..

[29]  Hong Wang,et al.  A fractional phase-field model using an infinitesimal generator of α stable Lévy process , 2019, J. Comput. Phys..

[30]  Jie Shen,et al.  A New Class of Efficient and Robust Energy Stable Schemes for Gradient Flows , 2017, SIAM Rev..

[31]  Jie Shen,et al.  Spectral Methods: Algorithms, Analysis and Applications , 2011 .

[32]  Jie Shen,et al.  Efficient Spectral Methods for PDEs with Spectral Fractional Laplacian , 2021, Journal of Scientific Computing.

[33]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[34]  Hong Wang,et al.  A space-time fractional phase-field model with tunable sharpness and decay behavior and its efficient numerical simulation , 2017, J. Comput. Phys..

[35]  Daniel J. Arrigo,et al.  An Introduction to Partial Differential Equations , 2017, An Introduction to Partial Differential Equations.

[36]  Hong Wang,et al.  Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation , 2018, Comput. Math. Appl..

[37]  Yanzhi Zhang,et al.  A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem , 2018, J. Comput. Phys..

[38]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[39]  Nicholas Hale,et al.  An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations , 2012, SIAM J. Sci. Comput..