Computational modeling of the initiation and propagation of complex fracture is central to the
discipline of engineering fracture mechanics. This review focuses on two promising …

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

The Cahn–Hilliard equation involves fourth-order spatial derivatives. Finite element
solutions are not common because primal variational formulations of fourth-order operators …

This paper develops and analyzes some fully discrete finite element methods for a parabolic
system consisting of the Navier--Stokes equations and the Cahn--Hilliard equation, which …

Liquid crystals are a type of soft matter that is intermediate between crystalline solids and
isotropic fluids. The study of liquid crystals has made tremendous progress over the past four …

A discontinuous Galerkin finite element method has been developed to treat the high-order
spatial derivatives appearing in the Cahn–Hilliard equation. The Cahn–Hilliard equation is a …

We propose and analyze a semi-discrete and a fully discrete mixed finite element method for
the Cahn-Hilliard equation ut+ Δ (ɛ Δ u− ɛ− 1 f (u))= 0, where ɛ> 0 is a small parameter …

The numerical approximations to the Allen–Cahn type diffuse interface models are studied,
with a particular focus on their performance in the sharp interface limit and the effectiveness …

In this paper, we develop local discontinuous Galerkin (LDG) methods for the fourth order
nonlinear Cahn–Hilliard equation and system. The energy stability of the LDG methods is …

This paper presents a differential geometry based model for the analysis and computation of
the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to …