344 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 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
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
Analysis of a Darcy-Cahn-Hilliard Diffuse Interface Model for the Hele-Shaw Flow and its Fully Discrete Finite Element Approximation
In this paper we present PDE and finite element analyses for a system of
partial differential equations (PDEs) consisting of the Darcy equation and the
Cahn-Hilliard equation, which arises as a diffuse interface model for the two
phase Hele-Shaw flow. We propose a fully discrete implicit finite element
method for approximating the PDE system, which consists of the implicit Euler
method combined with a convex splitting energy strategy for the temporal
discretization, the standard finite element discretization for the pressure and
a split (or mixed) finite element discretization for the fourth order
Cahn-Hilliard equation. It is shown that the proposed numerical method
satisfies a mass conservation law in addition to a discrete energy law that
mimics the basic energy law for the Darcy-Cahn-Hilliard phase field model and
holds uniformly in the phase field parameter . With help of the
discrete energy law, we first prove that the fully discrete finite method is
unconditionally energy stable and uniquely solvable at each time step. We then
show that, using the compactness method, the finite element solution has an
accumulation point that is a weak solution of the PDE system. As a result, the
convergence result also provides a constructive proof of the existence of
global-in-time weak solutions to the Darcy-Cahn-Hilliard phase field model in
both two and three dimensions. Finally, we propose a nonlinear multigrid
iterative algorithm to solve the finite element equations at each time step.
Numerical experiments based on the overall solution method of combining the
proposed finite element discretization and the nonlinear multigrid solver are
presented to validate the theoretical results and to show the effectiveness of
the proposed fully discrete finite element method for approximating the
Darcy-Cahn-Hilliard phase field model.Comment: 30 pages, 4 tables, 2 figure
Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential
We present and analyze finite difference numerical schemes for the Allen
Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential.
Both the first order and second order accurate temporal algorithms are
considered. In the first order scheme, we treat the nonlinear logarithmic terms
and the surface diffusion term implicitly, and update the linear expansive term
and the mobility explicitly. We provide a theoretical justification that, this
numerical algorithm has a unique solution such that the positivity is always
preserved for the logarithmic arguments. In particular, our analysis reveals a
subtle fact: the singular nature of the logarithmic term around the values of
and 1 prevents the numerical solution reaching these singular values, so
that the numerical scheme is always well-defined as long as the numerical
solution stays similarly bounded at the previous time step. Furthermore, an
unconditional energy stability of the numerical scheme is derived, without any
restriction for the time step size. The unique solvability and the
positivity-preserving property for the second order scheme are proved using
similar ideas, in which the singular nature of the logarithmic term plays an
essential role. For both the first and second order accurate schemes, we are
able to derive an optimal rate convergence analysis, which gives the full order
error estimate. The case with a non-constant mobility is analyzed as well. We
also describe a practical and efficient multigrid solver for the proposed
numerical schemes, and present some numerical results, which demonstrate the
robustness of the numerical schemes
Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements
We consider the e�cient solution of the Cahn-Hilliard variational inequality using an implicit time discretization, which is formulated as an optimal control problem with pointwise constraints on the control. By applying a semi-smooth Newton method combined with a Moreau-Yosida regularization technique for handling the control constraints we show superlinear convergence in function space. At the heart of this method lies the solution of large and sparse linear systems for which we propose the use of preconditioned Krylov subspace solvers using an e�ective Schur complement approximation. Numerical results illustrate the competitiveness of this approach
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