A two-level finite element method for the Allen–Cahn equation

ABSTRACT We consider the fully implicit treatment for the nonlinear term of the Allen–Cahn equation. To solve the nonlinear problem efficiently, the two-level scheme is employed. We obtain the discrete energy law of the fully implicit scheme and two-level scheme with finite element method. Also, the convergence of the two-level method is presented. Finally, some numerical experiments are provided to confirm the theoretical analysis.

[1]  Yanren Hou,et al.  A two-level finite element method for the Navier–Stokes equations based on a new projection ☆ , 2010 .

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

[3]  Chert,et al.  Applications of semi-implicit Fourier-spectral method to phase field equations , 2004 .

[4]  Yueqiang Shang,et al.  A two-level subgrid stabilized Oseen iterative method for the steady Navier-Stokes equations , 2013, J. Comput. Phys..

[5]  Weiwei Sun,et al.  Stability and Convergence of the Crank-Nicolson/Adams-Bashforth scheme for the Time-Dependent Navier-Stokes Equations , 2007, SIAM J. Numer. Anal..

[6]  王志恒,et al.  Two-level consistent splitting methods based on three corrections for the time-dependent Navier-Stokes equations , 2016 .

[7]  Yinnian He,et al.  Some iterative finite element methods for steady Navier-Stokes equations with different viscosities , 2013, J. Comput. Phys..

[8]  Yanren Hou,et al.  Two-level methods for the Cahn-Hilliard equation , 2016, Math. Comput. Simul..

[9]  Lutz Tobiska,et al.  A Two-Level Method with Backtracking for the Navier--Stokes Equations , 1998 .

[10]  Yinnian He,et al.  A multilevel finite element method in space‐time for the Navier‐Stokes problem , 2005 .

[11]  Qiang Du,et al.  Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions , 2006, J. Comput. Phys..

[12]  Yinnian He,et al.  Two-Level Method Based on Finite Element and Crank-Nicolson Extrapolation for the Time-Dependent Navier-Stokes Equations , 2003, SIAM J. Numer. Anal..

[13]  Yanren Hou,et al.  An AIM and one-step Newton method for the Navier–Stokes equations☆ , 2001 .

[14]  Jinchao Xu.  Two-grid Discretization Techniques for Linear and Nonlinear PDEs , 1996 .

[15]  Yanren Hou,et al.  A Small Eddy Correction Method for Nonlinear Dissipative Evolutionary Equations , 2003, SIAM J. Numer. Anal..

[16]  Yinnian He,et al.  Multi-level spectral Galerkin method for the Navier–Stokes equations, II: time discretization , 2006, Adv. Comput. Math..

[17]  J. B. Burie,et al.  Multilevel Methods in Space and Time for the Navier--Stokes Equations , 1997 .

[18]  Jinchao Xu,et al.  Error estimates on a new nonlinear Galerkin method based on two-grid finite elements , 1995 .

[19]  Yanren Hou,et al.  A postprocessing mixed finite element method for the Navier–Stokes equations , 2009 .

[20]  Blanca Ayuso de Dios,et al.  The Postprocessed Mixed Finite-Element Method for the Navier-Stokes Equations , 2005, SIAM J. Numer. Anal..

[21]  Xinlong Feng,et al.  An efficient two-step algorithm for the stationary incompressible magnetohydrodynamic equations , 2017, Appl. Math. Comput..

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

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

[24]  Yinnian He,et al.  The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data , 2008, Math. Comput..

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

[26]  R. Nicolaides,et al.  Numerical analysis of a continuum model of phase transition , 1991 .

[27]  WangZhiheng,et al.  Two-level methods for the Cahn-Hilliard equation , 2016 .

[28]  R. Rannacher,et al.  Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization , 1990 .

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

[30]  Yanren Hou,et al.  A two-level consistent splitting scheme for the navier-stokes equations , 2016 .