MATH 273A: Scientific Computation
Fall quarter, 2005
TOPICS IN APPLIED MATHEMATICS AND COMPUTATIONAL SCIENCE
Instructor: Bo Li
A Tentative List of Topics
(to be partially covered)

1. Multiscale Methods

The quasicontinuum method for defects in crystalline solids.

Examples of the heterogeneous multiscale method:
stiff systems of ordinary differential equations for chemical reactions;
homogenization for composites and flow in a porous medium;
combined molecular dynamics and continuum simulation of solids.

Coupling molecular dynamics and continuum simulations, boundary conditions.
Application to the contact line problem in fluids,
from noslip to Navier to generalized Navier boundary conditions,
negative kinetic constants.
 Coupling the quantum level firstprinciple to molecular dynamics to continuum mechanics
simulations: the need? any hope? strategy? coupling? interface
conditions? implementation? and the mathematics?

2. Interface Dynamics

Examples: geometric motions;
solidification; epitaxial growth of thin films; microstructural evolution;
twophase flow; and motion of biomolecules.

Sharp interface models. The fronttracking method, application to the
surface evolution of solid films with dislocations.
The levelset method: a simple example.
A finiteelement levelset method for the stressdriven interface motion.

Phasefield models and numerical methods, time stepping, stability
beyond the Gronwall inequality, threshold dynamics.
3. Energy Minimization

Concept of free energy, basics of thermodynamics, the second law.

Examples: nonlinear elasticity; the GinzburgLandau functional for superconductivity;
the CahnHilliard functional for phase separation;
the PoissonBoltzmann model; the Helfrich membrane energy;
and the densityfunctional theory.

Weak convergence methods, Gammalimits as effective energies,
energies of martensitic thin films.

Variational methods for coarsening in gradient systems,
application to the coarsening in epitaxial growth of thin films with or without slope selection.

The PoissonBoltzmann model and its improvement, boundaryvalue problems.

The basics of the densityfunctional theory, the KohnSham equations, realspace calculations
using parallel adaptive finiteelement methods.
Last updated by Bo Li on October 17, 2005.