The stabilized semi-implicit finite element method for the surface Allen-Cahn equation

Two semi-implicit numerical methods are proposed for solving the surface Allen-Cahn equation which is a general mathematical model to describe phase separation on general surfaces. The spatial discretization is based on surface finite element method while the temporal discretization methods are first-and second-order stabilized semi-implicit schemes to guarantee the energy decay. The stability analysis and error estimate are provided for the stabilized semi-implicit schemes. Furthermore, the first-and second-order operator splitting methods are presented to compare with stabilized semi-implicit schemes. Some numerical experiments including phase separation and mean curvature flow on surfaces are performed to illustrate stability and accuracy of these methods.

[1]  H. Holden,et al.  Splitting methods for partial differential equations with rough solutions : analysis and MATLAB programs , 2010 .

[2]  Charles M. Elliott,et al.  Evolving surface finite element method for the Cahn–Hilliard equation , 2013, Numerische Mathematik.

[3]  Feng Qiu,et al.  Phase separation patterns for diblock copolymers on spherical surfaces: a finite volume method. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Xiaofeng Yang,et al.  Numerical approximations of Allen-Cahn and Cahn-Hilliard equations , 2010 .

[5]  Steven J. Ruuth,et al.  A simple embedding method for solving partial differential equations on surfaces , 2008, J. Comput. Phys..

[6]  Kinetics of Phase Ordering on Curved Surfaces , 1998, cond-mat/9804237.

[7]  Cheng Wang,et al.  Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations , 2014, J. Comput. Phys..

[8]  Charles M. Elliott,et al.  An h-narrow band finite-element method for elliptic equations on implicit surfaces , 2010 .

[9]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[10]  G. Dziuk.  Finite Elements for the Beltrami operator on arbitrary surfaces , 1988 .

[11]  Charles M. Elliott,et al.  Finite element methods for surface PDEs* , 2013, Acta Numerica.

[12]  Qiang Du,et al.  Finite element approximation of the Cahn–Hilliard equation on surfaces , 2011 .

[13]  Jie Shen,et al.  On the maximum principle preserving schemes for the generalized Allen–Cahn equation , 2016 .

[14]  T. Tang,et al.  Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation , 2013 .

[15]  Bengt Fornberg,et al.  Solving PDEs with radial basis functions * , 2015, Acta Numerica.

[16]  Tao Tang,et al.  Long Time Numerical Simulations for Phase-Field Problems Using p-Adaptive Spectral Deferred Correction Methods , 2015, SIAM J. Sci. Comput..

[17]  P. Souganidis,et al.  Phase Transitions and Generalized Motion by Mean Curvature , 1992 .

[18]  Xiaofeng Yang,et al.  Error analysis of stabilized semi-implicit method of Allen-Cahnequation , 2009 .

[19]  Davide Marenduzzo,et al.  Phase separation dynamics on curved surfaces , 2013 .

[20]  James A. Warren,et al.  An efficient algorithm for solving the phase field crystal model , 2008, J. Comput. Phys..

[21]  Xinlong Feng,et al.  A highly efficient operator-splitting finite element method for 2D/3D nonlinear Allen–Cahn equation , 2017 .

[22]  Jie Shen,et al.  Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy , 2012, SIAM J. Numer. Anal..

[23]  Cécile Piret,et al.  The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces , 2012, J. Comput. Phys..

[24]  Seunggyu Lee,et al.  Motion by mean curvature of curves on surfaces using the Allen–Cahn equation , 2015 .

[25]  Steven M. Wise,et al.  Unconditionally stable schemes for equations of thin film epitaxy , 2010 .

[26]  Xinlong Feng,et al.  Stabilized Crank-Nicolson/Adams-Bashforth Schemes for Phase Field Models , 2013 .

[27]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[28]  R. Desai,et al.  Phase-ordering kinetics on curved surfaces , 1997 .

[29]  Long-Qing Chen.  Phase-Field Models for Microstructure Evolution , 2002 .