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

Chen, Wenbin; Liu, Yuan; Wang, Cheng; Wise, Steven

doi:10.1090/mcom3052

Convergence analysis of a fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation
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by Wenbin Chen, Yuan Liu, Cheng Wang and Steven M. Wise PDF

Math. Comp. 85 (2016), 2231-2257 Request permission

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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