The global dynamics of discrete semilinear parabolic equations
SIAM Journal on Numerical Analysis
Multigrid
Conservative multigrid methods for Cahn-Hilliard fluids
Journal of Computational Physics
A multigrid finite element solver for the Cahn-Hilliard equation
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Efficient Numerical Solution of Cahn-Hilliard-Navier-Stokes Fluids in 2D
SIAM Journal on Scientific Computing
Journal of Computational Physics
An Energy-Stable and Convergent Finite-Difference Scheme for the Phase Field Crystal Equation
SIAM Journal on Numerical Analysis
An adaptive multigrid algorithm for simulating solid tumor growth using mixture models
Mathematical and Computer Modelling: An International Journal
Inpainting of Binary Images Using the Cahn–Hilliard Equation
IEEE Transactions on Image Processing
SIAM Journal on Numerical Analysis
An adaptive multigrid algorithm for simulating solid tumor growth using mixture models
Mathematical and Computer Modelling: An International Journal
On linear schemes for a Cahn-Hilliard diffuse interface model
Journal of Computational Physics
An adaptive time-stepping strategy for solving the phase field crystal model
Journal of Computational Physics
Journal of Computational Physics
Efficient Solvers of Discontinuous Galerkin Discretization for the Cahn---Hilliard Equations
Journal of Scientific Computing
Journal of Computational Physics
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We present an unconditionally energy stable and solvable finite difference scheme for the Cahn-Hilliard-Hele-Shaw (CHHS) equations, which arise in models for spinodal decomposition of a binary fluid in a Hele-Shaw cell, tumor growth and cell sorting, and two phase flows in porous media. We show that the CHHS system is a specialized conserved gradient-flow with respect to the usual Cahn-Hilliard (CH) energy, and thus techniques for bistable gradient equations are applicable. In particular, the scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step-size. Owing to energy stability, we show that the scheme is stable in the $L_{s}^{\infty}(0,T;H_{h}^{1})$ norm, and, assuming two spatial dimensions, we show in an appendix that the scheme is also stable in the $L_{s}^{2}(0,T;H_{h}^{2})$ norm. We demonstrate an efficient, practical nonlinear multigrid method for solving the equations. In particular, we provide evidence that the solver has nearly optimal complexity. We also include a convergence test that suggests that the global error is of first order in time and of second order in space.