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Convergence analysis of a fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation


Authors: Wenbin Chen, Yuan Liu, Cheng Wang and Steven M. Wise
Journal: Math. Comp. 85 (2016), 2231-2257
MSC (2010): Primary 65M06, 65M12, 35K55, 76D05
DOI: https://doi.org/10.1090/mcom3052
Published electronically: December 14, 2015
MathSciNet review: 3511281
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Abstract: We present an error analysis for an unconditionally energy stable, fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation, a modified Cahn-Hilliard equation coupled with the Darcy flow law. The scheme, proposed by S. M. Wise, is based on the idea of convex splitting. In this paper, we rigorously prove first order convergence in time and second order convergence in space. Instead of the (discrete) $ L_s^\infty (0,T;L_h^2) \cap L_s^2 (0,T; H_h^2)$ error estimate, which would represent the typical approach, we provide a discrete $ L_s^\infty (0,T; H_h^1) \cap L_s^2 (0,T; H_h^3 )$ error estimate for the phase variable, which allows us to treat the nonlinear convection term in a straightforward way. Our convergence is unconditional in the sense that the time step $ s$ is in no way constrained by the mesh spacing $ h$. This is accomplished with the help of an $ L_s^2 (0,T;H_h^3)$ bound of the numerical approximation of the phase variable. To facilitate both the stability and convergence analyses, we establish a finite difference analog of a Gagliardo-Nirenberg type inequality.


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Additional Information

Wenbin Chen
Affiliation: Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China 200433
Email: wbchen@fudan.edu.cn

Yuan Liu
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China 200433
Email: 12110180072@fudan.edu.cn

Cheng Wang
Affiliation: Department of Mathematics, University of Massachusetts, North Dartmouth, Massachusetts 02747
Email: cwang1@umassd.edu

Steven M. Wise
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Email: swise1@utk.edu

DOI: https://doi.org/10.1090/mcom3052
Keywords: Cahn-Hilliard-Hele-Shaw, Darcy's law, convex splitting, finite difference method, unconditional energy stability, discrete Gagliardo-Nirenberg inequality, discrete Gronwall inequality
Received by editor(s): March 24, 2014
Received by editor(s) in revised form: November 19, 2014, and February 23, 2015
Published electronically: December 14, 2015
Additional Notes: The third author is the corresponding author
Article copyright: © Copyright 2015 American Mathematical Society