An efficient nonconforming finite element two-grid method for Allen-Cahn equation

Abstract In this paper, superconvergent analysis of an efficient two-grid method is discussed for the Allen–Cahn equation with the nonconforming E Q 1 r o t finite element. The unconditional stability of the numerical scheme is proved based on the monotonically increasing character of the nonlinear term in the problem, while the previous works always require some certain stability conditions. By use of the typical properties of E Q 1 r o t element together with a more accurate estimate on the nonlinear term, the superclose result of order O ( h 2 + H 4 + τ ) in the broken H 1 -norm is deduced rigorously without the above restrictions for the first time. Furthermore, the global superconvergence behavior is derived through interpolated postprocessing skill. Numerical results illustrate that the proposed method is actually effective.

[1]  Wei Liu,et al.  A two-grid algorithm based on expanded mixed element discretizations for strongly nonlinear elliptic equations , 2014, Numerical Algorithms.

[2]  Ke Zhang,et al.  A two-level finite element method for the Allen–Cahn equation , 2019, Int. J. Comput. Math..

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

[4]  Shi,et al.  CONSTRAINED QUADRILATERAL NONCONFORMING ROTATED Q1 ELEMENT , 2005 .

[5]  Guangzhi Du,et al.  A parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem , 2017, Numerical Algorithms.

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

[7]  Jincheng Ren,et al.  Nonconforming mixed finite element approximation to the stationary Navier–Stokes equations on anisotropic meshes , 2009 .

[8]  Fei Liu,et al.  Stabilized semi‐implicit spectral deferred correction methods for Allen–Cahn and Cahn–Hilliard equations , 2015 .

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

[10]  Dongyang Shi,et al.  APPROXIMATION OF NONCONFORMING QUASI-WILSON ELEMENT FOR SINE-GORDON EQUATIONS * , 2013 .

[11]  Shao-chunChen,et al.  AN ANISOTROPIC NONCONFORMING FINITE ELEMENT WITH SOME SUPERCONVERGENCE RESULTS , 2005 .

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

[13]  Haijun Wu,et al.  A Posteriori Error Estimates and an Adaptive Finite Element Method for the Allen–Cahn Equation and the Mean Curvature Flow , 2005, J. Sci. Comput..

[14]  Zhifeng Weng,et al.  Fast explicit operator splitting method and time-step adaptivity for fractional non-local Allen–Cahn model ☆ , 2016 .

[15]  Yongho Choi,et al.  A finite difference method for a conservative Allen-Cahn equation on non-flat surfaces , 2017, J. Comput. Phys..

[16]  Jinchao Xu,et al.  A Novel Two-Grid Method for Semilinear Elliptic Equations , 1994, SIAM J. Sci. Comput..

[17]  Dongyang Shi,et al.  Nonconforming quadrilateral finite element method for a class of nonlinear sine-Gordon equations , 2013, Appl. Math. Comput..

[18]  Yang Wang,et al.  A two-grid method for incompressible miscible displacement problems by mixed finite element and Eulerian–Lagrangian localized adjoint methods , 2018, Journal of Mathematical Analysis and Applications.

[19]  Dongyang Shi,et al.  Superconvergence analysis of a two-grid method for semilinear parabolic equations , 2018, Appl. Math. Lett..