12 research outputs found
A Robust Solver for a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation
We develop a robust solver for a second order mixed finite element splitting
scheme for the Cahn-Hilliard equation. This work is an extension of our
previous work in which we developed a robust solver for a first order mixed
finite element splitting scheme for the Cahn-Hilliard equaion. The key
ingredient of the solver is a preconditioned minimal residual algorithm (with a
multigrid preconditioner) whose performance is independent of the spacial mesh
size and the time step size for a given interfacial width parameter. The
dependence on the interfacial width parameter is also mild.Comment: 17 pages, 3 figures, 4 tables. arXiv admin note: substantial text
overlap with arXiv:1709.0400
Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Method for the Cahn-Hilliard-Navier-Stokes System
In this paper, we present a novel second order in time mixed finite element
scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities.
The scheme combines a standard second order Crank-Nicholson method for the
Navier-Stokes equations and a modification to the Crank-Nicholson method for
the Cahn-Hilliard equation. In particular, a second order Adams-Bashforth
extrapolation and a trapezoidal rule are included to help preserve the energy
stability natural to the Cahn-Hilliard equation. We show that our scheme is
unconditionally energy stable with respect to a modification of the continuous
free energy of the PDE system. Specifically, the discrete phase variable is
shown to be bounded in and the discrete
chemical potential bounded in , for any time
and space step sizes, in two and three dimensions, and for any finite final
time . We subsequently prove that these variables along with the fluid
velocity converge with optimal rates in the appropriate energy norms in both
two and three dimensions.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1411.524
A robust solver for a second order mixed finite element method for the Cahn–Hilliard equation
We develop a robust solver for a second order mixed finite element splitting scheme for the Cahn–Hilliard equation. This work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the Cahn–Hilliard equation. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spatial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild
Numerical Analysis of Convex Splitting Schemes for Cahn-Hilliard and Coupled Cahn-Hilliard-Fluid-Flow Equations
This dissertation investigates numerical schemes for the Cahn-Hilliard equation and the Cahn-Hilliard equation coupled with a Darcy-Stokes flow. Considered independently, the Cahn-Hilliard equation is a model for spinodal decomposition and domain coarsening. When coupled with a Darcy-Stokes flow, the resulting system describes the flow of a very viscous block copolymer fluid. Challenges in creating numerical schemes for these equations arise due to the nonlinear nature and high derivative order of the Cahn-Hilliard equation. Further challenges arise during the coupling process as the coupling terms tend to be nonlinear as well. The numerical schemes presented herein preserve the energy dissipative structure of the Cahn- Hilliard equation while maintaining unique solvability and optimal error bounds.
Specifically, we devise and analyze two mixed finite element schemes: a first order in time numerical scheme for a modified Cahn-Hilliard equation coupled with a non- steady Darcy-Stokes flow and a second order in time numerical scheme for the Cahn- Hilliard equation in two and three dimensions. The time discretizations are based on a convex splitting of the energy of the systems. We prove that our schemes are unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energies and unconditionally uniquely solvable. For each system, we prove that the discrete phase variable is essentially bounded in both time and space with respect to the Lebesque integral and the discrete chemical potential is Lesbegue square integrable in space and essentially bounded in time. We show these bounds are completely independent of the time and space step sizes in two and three dimensions. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms. The analyses included in this dissertation will provide a bridge to the development of stable, efficient, and optimally convergent numerical schemes for more robust and descriptive coupled Cahn-Hilliard-Fluid-Flow systems