High Order Finite Element Calculations for the Cahn-Hilliard Equation

  title={High Order Finite Element Calculations for the Cahn-Hilliard Equation},
  author={Ludovic Gouden{\`e}ge and Daniel Martin and Gr{\'e}gory Vial},
  journal={Journal of Scientific Computing},
In this work, we propose a numerical method based on high degree continuous nodal elements for the Cahn-Hilliard evolution. The use of the p-version of the finite element method proves to be very efficient avoiding difficult computations or strategies like $\mathcal{C}^{1}$ elements, adaptive mesh refinement, multi-grid resolution or isogeometric analysis. Beyond the classical benchmarks and comparisons with other existing methods, a numerical study has been carried out to investigate the… 
A Reduced Order Model for a stable embedded boundary parametrized Cahn-Hilliard phase-field system based on cut finite elements
This work investigates for the first time with a cut finite element method, a parameterized fourth-order nonlinear geometrical PDE, namely the Cahn-Hilliard system, and manages to find an efficient global, concerning the geometric manifold, and independent of geometric changes, reduced-order basis.
Higher order spectral element scheme for two- and three-dimensional Cahn–Hilliard equation
AbstractHigher order spectral element scheme is presented for Cahn–Hilliard equation in two and three dimensions. Legendre polynomial based nodal spectral element method is employed in space whereas
An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations
We consider the initial-boundary value problem for the 3D regularized compressible isothermal Navier–Stokes–Cahn–Hilliard equations describing flows of a two-component two-phase mixture taking into
65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial- boundary value problems 35Kxx Parabolic equations and parabolic systems 65Nxx Numerical methods for partial differential equations, boundary value problems
  • Mathematics
  • 2021
In the present work, we investigate a cut finite element method for the parameterized system of second-order equations stemming from the splitting approach of a fourth order nonlinear geometrical
Convergence to equilibrium for a second-order time semi-discretization ofthe Cahn-Hilliard equation
We consider a second-order two-step time semi-discretization of the Cahn-Hilliard equation with an analytic nonlinearity. The time-step is chosen small enough so that the pseudo-energy associated
Simulation of SPDE's for Excitable Media using Finite Elements
In this paper, we address the question of the discretization of Stochastic Partial Differential Equations (SPDE's) for excitable media. Working with SPDE's driven by colored noise, we consider a
Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation
We review space and time discretizations of the Cahn-Hilliard equation which are energy stable. In many cases, we prove that a solution converges to a steady state as time goes to infinity. The proof


A multigrid finite element solver for the Cahn-Hilliard equation
Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy
SummaryA fully discrete finite element method for the Cahn-Hilliard equation with a logarithmic free energy based on the backward Euler method is analysed. Existence and uniqueness of the numerical
Approximation of Cahn–Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements
A variational formulation and C1 finite element scheme with adaptive mesh refinement and coarsening are developed for phase‐separation processes described by the Cahn–Hilliard diffuse interface model
Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition
Fully discrete discontinuous Galerkin methods with variable meshes in time are developed for the fourth order Cahn-Hilliard equation arising from phase transition in materials science and are proved to give optimal order error bounds.
On the Convergence of the p-Version of the Boundary Element Galerkin Method.
Abstract : The authors consider various physical problems which may be formulated in terms of integral equations of the first kind, including the two-dimensional screen Neumann and Dirichlet problems
Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility
It is proved that there exists a unique solution for sufficiently smooth initial data in the Cahn-Hilliard equation and an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions is proved.