A Splitting Method for the Cahn–hilliard Equation With Inertial Term

Abstract

We prove that the semidiscrete solution converges weakly to the continuous solution as the discretization parameter tends to 0. We obtain optimal a priori error estimates, assuming enough regularity on the solution. We also show that the semidiscrete solution converges to an equilibrium as time goes to infinity and we give a simple finite difference version of the scheme.

On the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard Equation with Logarithmic Nonlinear Terms

“…which has the advantage of splitting the fourth-order (in space) equation into a system of two second-order ones (see [16], [18] and [19]). Consequently, we use a P1-finite element for the space discretization, together with a semi-implicit Euler time discretization (i.e., implicit for the linear terms and explicit for the nonlinear ones).…”

Energy stable and convergent finite element schemes for the modified phase field crystal equation

“…The main purpose of this paper is to derive and analyze a second order (in time) fully discrete finite element scheme for the MPFC equation. For the space discretization we use a splitting approach which is well known in the context of phase field models (cf., e.g., [9,21]). This argument allows to consider low order (piecewise linear) finite elements, although the analysis is carried out in a more general setting, namely a Galerkin approximation.…”

The Cahn-Hilliard Equation with Logarithmic Potentials

“…We consider here the scheme introduced in [79] (see also [50], [126] and [135]) which has the advantage of splitting the fourth-order equation into a system of two secondorder ones. Consequently, we use a P 1 -finite element for the space discretization, together with a semi-implicit Euler time discretization (i.e., implicit for the linear terms and explicit for the nonlinear ones).…”

“…Cahn-Hilliard Equation 567 q [126], [134], [135], [136], [138], [139], [141], [142], [143], [144], [145], [146], [147], [152], [158], [191], [200], [201], [203], [208], [209], [211] and [217] for the numerical analysis and simulations of the Cahn-Hilliard equation (and several of its generalizations).…”