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Numerical Algorithms
Published in: Numerical Algorithms 2/2020

12-11-2019 | Original Paper

Superconvergence analysis of a two-grid method for an energy-stable Ciarlet-Raviart type scheme of Cahn-Hilliard equation

Authors: Qian Liu, Dongyang Shi

Published in: Numerical Algorithms | Issue 2/2020

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Abstract

In this paper, superconvergence analysis of a mixed finite element method (MFEM) combined with the two-grid method (TGM) is presented for the Cahn-Hilliard (CH) equation for the first time. In particular, the discrete energy-stable Ciarlet-Raviart scheme is constructed with the bilinear element. By use of the high accuracy character of the element, the superclose estimates are deduced for both of the traditional MFEM and of the TGM. Crucially, the main difficulty brought by the coupling of the unknowns is dealt with by some technical methods. Furthermore, the global superconvergent results are achieved by interpolation postprocessing skill. Numerical results illustrate that the proposed TGM is very effective and its computing cost is almost one-third of that of the traditional FEM without loss of accuracy.
previous article Rounding error analysis of divided differences schemes: Newton’s divided differences; Neville’s algorithm; Richardson extrapolation; Romberg quadrature; etc.
next article Numerical methods for static shallow shells lying over an obstacle

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Literature
1.
2.
Cahn, J.W., Hilliard, JE: Free energy of a nonuniform system I, Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)MATHCrossRef
3.
Hilliard, J.E.: Spinodal decomposition. Phase transformations in ASM Cleveland, 497–560 (1970)
4.
Novick, C.A.: The Cahn-Hilliard equation:mathematical and modeling perspectives. Adv. Math. Sci. Appl. 8(2), 965–985 (1998)MathSciNetMATH
5.
Cahn, J.W., Elliot, C.M., Novick-Cohen, A.: The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. Eur. J. Appl. Math. 7(3), 287–301 (1996)MathSciNetMATHCrossRef
6.
Cahn, J.W., Novick, C.A.: Limiting motion for anAllen-Cahn/Cahn-Hilliard system. In: Free Boundary Problems Theory and Applications, pp 89–97, Harlow (1996)
7.
8.
Miranville, A., Zelik, S.: Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions. Math. Methods Appl. Sci. 28, 709–735 (2005)MathSciNetMATHCrossRef
9.
Zheng, S., Milani, A.: Global attractors for singular perturbations of the Cahn-Hilliard equations. J. Diff. Equations 209, 101–139 (2005)MathSciNetMATHCrossRef
10.
Willie, R.: The Cahn-Hilliard equation with boundary nonlinearity and high viscosity. Methods Appl. Anal. 10, 589–602 (2003)MathSciNetMATH
11.
Alikakos, N.D., Fusco, G., Karali, G.: Motion of bubbles towards the boundary for the Cahn-Hilliard equation. Euro. J. Appl. Math. 15, 103–124 (2004)MathSciNetMATHCrossRef
12.
He, Y., Liu, Y.: Stability and convergence of the spectral Galerkin method for the Cahn-Hilliard equation. Numer. Methods Partial Differ. Eq. 24(6), 1485–1500 (2008)MathSciNetMATHCrossRef
13.
Li, D., Qiao, Z.: On second order semi-implicit fourier spectral methods for 2D Cahn-Hilliard equations. J. Sci. Comput. 70(1), 301–34 (2017)MathSciNetMATHCrossRef
14.
Khiari, N., Achouri, T., Ben Mohamed, M.L., Omrani, K.: Finite difference approximate solutions for the Cahn-Hilliard equation. Numer. Methods Partial Differ. Eq. 23(2), 437–455 (2010)MathSciNetMATHCrossRef
15.
Chen, K., Feng, W., Wang, C., Wise, S.M.: An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation. J. Comput. Appl. Math. 362 (15), 574–595 (2019)MathSciNetMATHCrossRef
16.
Tang, P., Qiu, F., Zhang, H., Yang, Y.: Phase separation patterns for diblock copolymers on spherical surfaces: a finite volume method. Phys. Rev. E. 72, 016710 (2005)CrossRef
17.
Elliott, C.M., French, D.A.: A nonconforming finite-element method for the two dimensional Cahn-Hilliard equation. SIAM J. Numer. Anal. 26(4), 884–903 (1989)MathSciNetMATHCrossRef
18.
Liu, Q., Chen, L., Zhou, J.: Two-level methods for the Cahn-Hilliard equation. Math. Comput. in Simula. 126(C), 89–103 (2016)MathSciNetCrossRef
19.
Wang, W., Chen, L., Zhou, J.: Postprocessing mixed finite element methods for solving Cahn-Hilliard equation: methods and error analysis. J. Sci. Comput. 67, 724–746 (2016)MathSciNetMATHCrossRef
20.
Elliott, C.M., French, D.A.: Numerical studies of the Cahn-Hilliard equation for phase separation. IMA J. Appl. Math. 38(2), 97–128 (1987)MathSciNetMATHCrossRef
21.
Feng, X., Prohl, A.: Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99, 47–84 (2004)MathSciNetMATHCrossRef
22.
Kay, D., Styles, V., Süli, E.: Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection. SIAM J. Numer. Anal. 47(4), 2660–2685 (2009)MathSciNetMATHCrossRef
23.
Brenner, S.C., Diegel, A.E., Li-Yeng, S.: A robust solver for a mixed finite element method for the Cahn-Hilliard equation. J. Sci. Comput. 77, 1234–1249 (2018)MathSciNetMATHCrossRef
24.
Wu, X., van Zwieten, G.J., van der Zee, K.G.: Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models. Int. J. Numer. Methods Biomed. Eng. 30, 180–203 (2014)MathSciNetCrossRef
25.
Diegel, A.E., Wang, C., Wise, S.M.: Stability and convergence of a second order mixed finite element method for the Cahn-Hilliard equation. IMA J. Numer. AnaL. 36, 1867–1897 (2016)MathSciNetMATHCrossRef
26.
Yan, Y., Chen, W., Wang, C., Wise, S.M.: A second-order energy stable BDF numerical scheme for the Cahn-Hilliard equation. Commun. Comput. Phys. 23 (2), 572–602 (2018)MathSciNetCrossRef
27.
28.
29.
Dawson, C., Wheeler, M., Woodward, C.: A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. 35, 435–452 (1998)MathSciNetMATHCrossRef
30.
Chen, Y., Huang, Y., Yu, D.: A two-grid method for expanded mixed finite-element solution of semilinear reaction diffusion equations. Int. J. Numer. Mech. Eng. 57, 193–209 (2003)MathSciNetMATHCrossRef
31.
Liu, W., Yin, Z., Li, J.: A two-grid algorithm based on expanded mixed element discretizations for strongly nonlinear elliptic equations. Numer. Algor. 70(1), 93–111 (2015)MathSciNetMATHCrossRef
32.
Shi, D., Yang, H.: Unconditional optimal error estimates of a two-grid method for semilinear parabolic equation. Appl. Math. Comput. 310, 40–47 (2017)MathSciNetMATH
33.
Chen, Y., Chen, L., Zhang, X.: Two-grid method for nonlinear parabolic equations by expanded mixed finite element methods. Numer. Methods Partial Differ. Eq. 29(4), 1238–1256 (2013)MathSciNetMATHCrossRef
34.
Zuo, L., Du, G.: A parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem. Numer. Algor. 77(1), 151–165 (2018)MathSciNetMATHCrossRef
35.
Wang, Y., Chen, Y.: A two-grid method for incompressible miscible displacement problems by mixed finite element and Eulerian-Lagrangian localized adjoint methods. J. Math. Anal. Appl. 468, 406–422 (2018)MathSciNetMATHCrossRef
36.
Shi, D., Mu, P., Yang, H.: Superconvergence analysis of a two-grid method for semilinear parabolic equations. Appl. Math. Lett. 84, 34–41 (2018)MathSciNetMATHCrossRef
37.
Zhou, J., Chen, L., Huang, Y., Wang, W.: An efficient two-grid scheme for the Cahn-Hilliard equation. Commun. Comput. Phys. 17(1), 127–145 (2015)MathSciNetMATHCrossRef
38.
Lin, Q., Lin, J.: Finite Element Methods: Accuracy and Improvement. Science Press, Beijing (2006)
39.
Diegel, A., Feng, X., Wise, S.M.: Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system. SIAM J. Numer. Anal. 53(1), 127–152 (2015)MathSciNetMATHCrossRef
40.
Lin, Q., Tobiska, L., Zhou, A.: Superconvergence and extrapolation of nonconforming low order finite elements applied to the Poisson equation. IMA J. Numer. Anal. 25(1), 160–181 (2005)MathSciNetMATHCrossRef
41.
Shi, D., Pei, L.: Nonconforming quadrilateral finite element method for a class of nonlinear sine-Gordon equations. Appl. Math. Comput. 219(17), 9447–9460 (2013)MathSciNetMATH
Metadata
Title
Superconvergence analysis of a two-grid method for an energy-stable Ciarlet-Raviart type scheme of Cahn-Hilliard equation
Authors
Qian Liu
Dongyang Shi
Publication date
12-11-2019
Publisher
Springer US
Published in
Numerical Algorithms / Issue 2/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-019-00829-0

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