This chapter surveys recent numerical advances in the phase field method for geometric
surface evolution and related geometric nonlinear partial differential equations (PDEs) …

The celebrated Cahn–Hilliard (CH) equation was proposed to model the process of phase
separation in binary alloys by Cahn and Hilliard. Since then the equation has been …

In this paper we develop an evolution of the C^1 virtual elements of minimal degree for the
approximation of the Cahn--Hilliard equation. The proposed method has the advantage of …

We introduce provably unconditionally stable mixed variational methods for phase-field
models. Our formulation is based on a mixed finite element method for space discretization …

We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem
used to model microphase separation in diblock copolymer melts. We show that critical …

We consider the so-called Cahn–Hilliard–Oono equation with singular (eg logarithmic)
potential in a bounded domain of ℝ d, d≤ 3. The equation is subject to an initial condition …

In this study, we present a robust and efficient fingerprint image restoration algorithm using
the nonlocal Cahn–Hilliard (CH) equation, which was proposed for modeling the …

Using a framework of partial differential equation-constrained optimization, we demonstrate
that multiple constitutive relations can be extracted simultaneously from a small set of …

The first objective of this work is to present, to some extent, a deep introduction to the basic
concepts on real and functional analysis. In principle, the text is written for applied …

We propose new collocation methods for phase-field models. Our algorithms are based on
isogeometric analysis, a new technology that makes use of functions from computational …