AN ENERGY-STABLE PARAMETRICFINITE ELEMENT METHOD FOR SIMULATING SOLID-STATE DEWETTING PROBLEMS INTHREE DIMENSIONS

We propose an accurate and energy-stable parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting in three-dimensional space.The model describes the motion of the film/vapor interface with contact line migration and is governed by the surface diffusion equation with proper boundary conditions at the contact line.We present a weak formulation for the problem,in which the contact angle condition is weakly enforced.By using piecewise linear elements in space and backward Euler method in time,we then discretize the formulation to obtain a parametric finite element approximation,where the interface and its contact line are evolved simultaneously.The resulting numerical method is shown to be well-posed and unconditionally energystable.Furthermore,the numerical method is generalized to the case of anisotropic surface energies in the Riemannian metric form.Numerical results are reported to show the convergence and efficiency of the proposed numerical method as well as the anisotropic effects on the morphological evolution of thin films in solid-state dewetting.

The Kinematic Effects of the Defects in Liquid Crystal Dynamics

Here we investigate the kinematic transports of the defects in the nematic liquid crystal system by numerical experiments.The model is a shear flow case of the viscoelastic continuummodel simplified fromthe Ericksen-Leslie system.The numerical experiments are carried out by using a differencemethod.Based on these numerical experiments we find some interesting and important relationships between the kinematic transports and the characteristics of the flow.We present the development and interaction of the defects.These results are partly consistent with the observation from the experiments.Thus this scheme illustrates,to some extent,the kinematic effects of the defects.

Numerical Study of Quantized Vortex Interaction in theGinzburg-Landau Equation on Bounded Domains

In this paper,we study numerically quantized vortex dynamics and their interaction in the two-dimensional(2D)Ginzburg-Landau equation(GLE)with a dimensionless parameter#>0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition.We begin with a reviewof the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically.Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition.Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws,we simulate quantized vortex interaction of GLE with different#and under different initial setups including single vortex,vortex pair,vortex dipole and vortex lattice,compare them with those obtained from the corresponding reduced dynamical laws,and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction.Finally,we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.

A Uniformly Convergent Numerical Method for Singularly Perturbed Nonlinear Eigenvalue Problems

In this paper we propose a uniformly convergent numerical method for discretizing singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation and quantum chemistry.We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint.Matched asymptotic approximations for the problem are presented to locate the positions and characterize the widths of boundary layers and/or interior layers in the solution.A uniformly convergent numerical method is proposed by using the normalized gradient flow and piecewise uniform mesh techniques based on the asymptotic approximations for the problem.Extensive numerical results are reported to demonstrate the effectiveness of our numerical method for the problems.Finally,the method is applied to compute ground and excited states of Bose-Einstein condensation in the semiclassical regime and some conclusive findings are reported.

A Time-Splitting Spectral Method for Three-Wave Interactions in Media with Competing Quadratic and Cubic Nonlinearities

This paper introduces an extension of the time-splitting spectral(TSSP)method for solving a general model of three-wave optical interactions,which typically arises from nonlinear optics,when the transmission media has competing quadratic and cubic nonlinearities.The key idea is to formulate the terms related to quadratic and cubic nonlinearities into a Hermitian matrix in a proper way,which allows us to develop an explicit and unconditionally stable numerical method for the problem.Furthermore,the method is spectral accurate in transverse coordinates and second-order accurate in propagation direction,is time reversible and time transverse invariant,and conserves the total wave energy(or power or the norm of the solutions)in discretized level.Numerical examples are presented to demonstrate the efficiency and high resolution of the method.Finally the method is applied to study dynamics and interactions between three-wave solitons and continuous waves in media with competing quadratic and cubic nonlinearities in one dimension(1D)and 2D.