QD's research page

$Q. Du$
$Vita$ - $Research$ - $Teaching$ - $Personal$ - $Links$

Gateway to Qiang's research world

Research sponsored in part by NSF-DMS, NSF-CISE, NSF-DMR, NIH-NCI and NSF IUCRC

Publications may be cross listed below. For more up-to-date info, check the publication list by years
You are visitor $Counter$ Since Aug 21, 2001 (last updated Apr 2005)

Numerical solution of PDEs

Focusing on the design and analysis of algorithms
for PDEs in various applications; Discretizations
include finite element, finite difference/volume
& spectral methods. Research also includes mesh
generation/optimization, meshfree computation,
domain decomposition and parallelization

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The mathematical analysis of algorithms
centers on the study of well-posedness:
consistency, stability, convergence,
error estimates. Also interested in the
preservation of the physical, geometric
and/or topological structures/invariants.

Finite element methods

Analysis and Approximation of the Ginzburg-Landau Model of Superconductivity, SIAM Review, 34 54-81, 1992.
   one of the popular references on G-L models with description of the
   physical background as well as analytic and numerical techniques
   Search Ginzburg Landau Model on Google Scholar for relevant links
   (pdf file for download)
Numerical Analysis of a Continuum Model of Phase Transition, SIAM Numer. Anal., 28, pp 1310-1322, 1991.
a paper on finite element solution of Cahn-Hilliard model,
mostly 1d analysis, an unpublished tech report available for 2d
Solving the Ginzburg-Landau equations by finite-element methods , Phys Rev B,46, pp. 9027-9034, 1992
application of FEM to G-L models, numerics have been refined in later publications
Finite element approximation of a periodic Ginzburg-Landau model for type-II superconductors , Numer. Math., 64, 85-114, 1993.
convergence and error estimates for FEM of G-L models
A finite element domain decomposition method for parabolic equations Domain Decomposition Methods for PDEs, (SIAM) 255-263, 1991
description of the explicit/implicit coupling in the FEM setting, error estimates.
Numerical Algorithms of the Lawrence--Doniach Model for Layered Superconductors and their Parallel Implementation , SIAM Sci Comp 20, p2122-2139, 1999.
provides the convergence of layer based algorithm, discussed parallel implementation
Optimization Based Nonoverlapping Domain Decomposition Algorithms and Their Convergence , SIAM Num Anal, 39, pp1056-1077, 2001.
a new framework for software integration and multidisciplinary solvers
unified several existing algorithms, analysis of convergence
A Gradient Method Approach to Optimization-Based Multidisciplinary Simulations and Nonoverlapping Domain Decomposition Algorithms , SIAM Numer. Anal. 37, 1513-1541, 2000.
another way to solve software integration and multidisciplinary problems
Semidiscrete Finite Element Approximations of a Linear Fluid-Structure Interaction Problem , SIAM J. Numer. Anal., 42, pp.1-29, 2004.
approximation of Navier-Stokes + linear elasticity
A low dispersion, high accuracy finite element method for first order hyperbolic systems in several space variables , Comp. Math Appl., 1988
examines convergence, stability and other properties
Numerical Approximation of Some Linear Stochastic Partial Differential Equations Driven by Special Additive Noises , SIAM J. Numer. Anal., 40, pp.1421-1445, 2002.
convergence of approximation depends on type of noises
Finite-Element Approximations of a Ladyzhenskaya Model for Stationary Incompressible Viscous Flow, SIAM J. Numer. Anal. 27, 1-19, 1990.
when N-S breaks down, Ladyzhenskaya's model still gives good solution
Analysis and finite element approximation of optimal control problems for a Ladyzhenskaya model for stationary, incompressible, viscous flows , J. Comp. Appl. Math. 61, pp 323-343, 1995
then, might be easier to control Ladyzhenskaya's model
Numerical Solution of the Three-Dimensional Ginzburg-Landau Models Using Artificial Boundary, SIAM Numer. Anal.36, pp.1482-1506, 1999.
algorithms and analysis for the coupled G-L equation with Maxwell in whole space
Finite Element Methods with Matching and Nonmatching Meshes for Maxwell Equations with Discontinuous Coefficients, SIAM Numer. Anal.,37, pp.1542-1570, 2000.
domain decomposition for Maxwell equations
Mesh and solver co-adaptation in finite element methods for anisotropic problems , Numerical Methods for Differential Equations, 21, pp.859-874, 2005.
for maximum efficiency, should jointly adapt mesh and solver
On Mesh Geometry and Stiffness Matrix Conditioning for General Finite Element Spaces, SIAM J. Numer. Analy., 2009
What is the precise dependence of the stiffness matrix conditioning on the mesh geometry? We provide a comprehensive analysis.
Adaptive Finite Element Method for a Phase Field Bending Elasticity Model of Vesicle Membrane Deformations, 30, no3, pp.1634-1657, 2008
posteriori error based adaptive finite element methods
Analysis of a Mixed Finite Element Method for a Phase Field Bending Elasticity Model of Vesicle Membrane Deformation J. Computational Math., 24 (3), pp.265-280, 2006.
mixed finite element for high order variational problem
Convergence of numerical approximations to a phase field bending elasticity model of membrane deformations, Inter. J. Numer. Anal and Modeling, 4, pp.441-459, 2006
Galerkin approximation to phase field bending elasticity model

Finite difference methods

A finite difference domain decomposition algorithm for numerical solution of the heat equation, Math of Comp., 57, pp63-71, 1991
describes the explicit/implicit coupling, variable mesh/steps
Efficient Parallel Algorithms for Parabolic Problems, SIAM Numer Anal,39, pp.1469-1487, 2001
higher order explicit/implicit coupling, parallel efficiency
Discrete gauge invariant approximations of the G-L models, Math. Comp., 67, pp.965-986, 1998
analysis of a popular time-dependent G-L solver

Finite volume methods

Analysis and Convergence of a Covolume Approximation of the Ginzburg-Landau Model of Superconductivity, SIAM Numer Anal 35, pp1049-1072, 1998;
generalizing gauge invariant schemes to unstructured grids
Convergence Analysis of a Finite Volume Method for Maxwell's Equations in Nonhomogeneous Media , SIAM Numer Anal, 41, pp.37-63, 2003
co-volume based approximation, error estimates
Voronoi-based finite volume methods, optimal Voronoi meshes, and PDEs on the sphere Comput. Meth. Appl. Mech. Engrg. , 192, pp.3933-3957, 2003;
getting error estimates for FVM and demonstrate the superiority of SCVT
Finite Volume Methods on Spheres and Spherical Centroidal Voronoi Meshes ,SIAM J. Numer. Anal., 43, pp.1673-1692, 2005.
getting error estimates for FVM and demonstrate the superiority
Analysis of a mixed finite volume discretization of fourth-order equations on general surfaces, IMA Numerical Analysis, 2009.

Spectral methods

Spectral viscosity methods for multidimensional nonlinear conservation laws, Mathematics of Computation,61, 629-643, 1993
convergence of spectral viscosity methods, error estimates
Fourier Spectral Approximation to a Dissipative System Modeling the Flow of Liquid Crystals, SIAM J. Numer. Anal., 39, 735-762, 2001
apply Fourier spectral to NS coupled with director equation, analysis
Computing the Ground State Solution of Bose--Einstein Condensates by a Normalized Gradient Flow, SIAM J. Scientific Comp., 25, pp.1674-1697, 2004
coupled time-splitting and spectral methods for GP equations
The dynamics and interaction of quantized vortices in Ginzburg-Landau-Schroedinger equations SIAM J. Appl Math, 67, pp.1740-1775, 2007
hybrid spectral/finite element methods for GLS equations
Spectral Implementation of An Adaptive Moving Mesh Method for Phase-field Equations, J. Computational Phys., 220, pp.498-510, 2006.
Introducing adaptivity to spectral methods
A Fourier Spectral Moving Mesh Method for the Cahn-Hilliard Equation with Elasticity, Comm. in Computational Phys., 5, pp. 582-599, 2009
Generalizing moving mesh Fourier spectral to Cahn-Hilliard
Applications of Moving Mesh Methods to the Fourier Spectral Approximations of Phase-Field Equations, in Recent Advances in Computational Sciences: Selected Papers from the International Workshop on Computational Sciences and Its Education, edited by Jorgensen et. al., World Scientific, 2007 (ISBN: 981270700X)

Quantized vortices - superfluidity

Recent work on Bose-Einstein Condensate - the so-called fifth state of matter

(our computation on vortices in a BEC are remarkably close to experiments)

Vortices in a rotating Bose-Einstein condensate: Critical angular velocities and energy diagrams in the Thomas-Fermi regime , Phy Rev A, Dec 2001.
analyzing vortex state in BEC, similar to the G-L framework
numerical computations of vortex solutions, 2d

Dissipative Flow and Vortex Shedding in the Painleve Boundary Layer of a Bose-Einstein Condensate , Phy Rev Lett., 91 (2003), 090407
analyzing a recent MIT experiments with the laser probing of the BEC,
explains the onset of dissipation and its relation to vortex shedding

Computing the Ground State Solution of Bose--Einstein Condensates by a Normalized Gradient Flow, SIAM J. Scientific Comp., 25, pp.1674-1697, 2004.
developing algorithms for computing the ground state

Dynamics of Rotating Bose--Einstein Condensates and its Efficient and Accurate Numerical Computation, SIAM Applied Math. ,66, pp.758-786, 2006

The dynamics and interaction of quantized vortices in Ginzburg-La ndau-Schroedinger equations, SIAM J. Appl Math, 67, pp.1740-1775, 2007

Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrodinger equation, Euro. J. Appl. Math., 18, pp.607-630, 2007

Modeling, analysis and computations in superconductivity

Research in this direction has focused on the study of quantized vortices in superconductors, using mezoscale models as typified by the celebrated Ginzburg-Landau equations. Though we have also worked on the microscopic BCS models that can be used to understand the basic structure of superconductors and of the atomic and sub-atomic behavior of these materials, as well as the macroscopic vortex density and critical state models that can be of use for the design of devices. Mezoscale models are of great use in understanding important phenomena in superconductors such as vortex nucleation, motion and interaction, vortex pinning, critical fields and currents, Josephson effects, inhomogeneities and fluctuations, vortex glass and vortex fluid structures, resistivity, etc.
Our efforts in superconductivity center on the following aspects:

* to develop, refine and simplify mezoscale and macroscale models for superconductivity, via rigorous analysis, computation and asymptotic studies
* to analyze the models to gain further understanding of the properties of these models and of their solutions, and also to determine their validity and usefulness for solving physically interesting problems;
* to develop, analyze, and implement algorithms for the numerical simulation of solutions of the various models including algorithms that give gauge invariant approximations, those based on artificial bc, and those which are parallelizable
* to use our algorithms and codes to study superconducting phenomena.
(link to gallery)

Analysis and Approximation of the Ginzburg-Landau Model of Superconductivity , SIAM Review, 34, pp. 54-81, 1992.

Numerical method for BCS model, Lett. Math. Phys.

FEM for the periodic Ginzburg-Landau model, Numer. Math.

Modeling and Analysis of a Periodic Ginzburg-Landau Model for Type-II Superconductors, SIAM Appl Math., 53, pp. 689-717, 1993

Solving the Ginzburg-Landau equations by finite-element methods , Phys Rev B 46, pp. 9027-9034, 1992

Global Existence and uniqueness of solution of the time-dependent G-L equations in superconductivity, Applicable Anal. 53, pp. 1-17, 1994

FEM for time-dependent Ginzburg-Landau model, Comp. Math. Appl. 27, pp. 119-133, 1994

On the Lawrence-Doniach and Anisotropic Ginzburg-Landau Models for Layered Superconductors, SIAM Appl. Math. 55, pp. 156-174, 1995

High-Kappa Limits of the Time-Dependent Ginzburg-Landau Model, SIAM Appl. Math. 56, pp. 1060-1093, 1996

A limit G-L model for thin films with variable thickness, ZAMP, 47, pp410-431, 1996

Analysis and Convergence of a Covolume Approximation of the Ginzburg-Landau Model of Superconductivity, SIAM J. Numer. Anal. ,35, pp. 1049-1072, 1998

Ginzburg-Landau Vortices: Dynamics, Pinning, and Hysteresis, SIAM Math. Anal., 28, pp. 1265-1293, 1997

Vortices in superconductors: modelling and computer simulations Phil.Tran.Roy.Soc.London.A,355, pp1957-1968, 1997

Numerical Algorithms of the Lawrence--Doniach Model for Layered Superconductors and their Parallel Implementation SIAM Sci. Comp., 20, pp. 2122-2139, 1999

Discrete gauge invariant approximations of G-L model , Math Comp 67, pp.965-986, 1998

Simplified S-N-S junction models & their approximation, Euro J App Math10, pp.1-25, 1999

Computational simulation of type-II superconductivity including pinning phenomena Phys. Rev. B 1995

Numerical Solution of the Three-Dimensional Ginzburg-Landau Models Using Artificial Boundary, SIAM Num. Anal., 36, pp. 1482-1506, 1999

Studies of a Ginzburg-Landau Model for d-Wave Superconductors, SIAM Appl. Math., 59, pp. 1225-1250, 1999

Analysis and computation of a mean-field model for superconductivity Numerische Mathematik

Convergence Analysis of a Numerical Method for a Mean Field Model of Superconducting Vortices, SIAM Numer. Anal., 37, pp. 911-926, 2000

An upwinding mixed finite element method for a mean field model of superconducting vortices , Math. Modelling and Numer. Anal. 34 687-706, 2000

Existence of Weak Solutions to Some Vortex Density Models, SIAM Math. Anal., 34, pp. 1279-1299, 2003

Critical Magnetic Field and Asymptotic Behavior of Superconducting Thin Films, SIAM Math. Anal., 34, pp. 239-256, 2002

Stochastic dynamics of Ginzburg-Landau vortices in superconductors , Phys. Rev B pp.52506, 2001

Limiting Models for Josephson Junctions and Superconducting Weak Links , J. Math Anal Appl, 266, pp357-382, 2002

Bifurcation diagrams of Ginzburg-Landau model Physica D,163, pp.94-105, 2001

Modeling and Computation of Random Thermal Fluctuations and Material Defects in the Ginzburg-Landau Model for Superconductivity , J. Computational Physics, 181, 45-67, 2002

Numerical Simulations of the Quantized Vortices on a Thin Superconducting Hollow Sphere, with L. Ju, J. Comp. Phys., 201, pp.511-530, 2004

Approximations of a Ginzburg-Landau model for superconducting hollow spheres based on spherical centroidal Voronoi tessellations , Mathematics of Computation, 74, pp.1257-1280, 2005.

Existence of Weak Solutions to Some Vortex Density Models, SIAM J. Math. Anal., 34, pp. 1279-1299, 2003

Numerical approximations of the Ginzburg-Landau models for superconductivity , J. Mathematical Physics 46, pp.095109 1-22, 2005
A recent invited review on the topic.

Others (including recent ones) available upon request

or check out the complete listing

Link to the ISU superconductivity page

Link to the Paul's superconductivity movies

Computational Materials

(research supported in part by NSF DMS and NSF ITR)

Computation of
coarsening using
G-L eq. and ETD

Critical nuclei (link to gallery)

Phase field modeling

Numerical Analysis of a Continuum Model of Phase Transition, SIAM Numer. Anal., 28, pp 1310-1322, 1991
An earlier paper on the numerical solution of the Cahn-Hilliard equations satisfied energy dissipation and volume conservation.
A Phase Field Approach in the Numerical Study of the Elastic Bending Energy for Vesicle Membranes, J. of Comp. Phy., 198, pp450-468, 2004
provide a framework to model bending energy within the diffuse interface approach
Simulating the Deformation of Vesicle Membranes under Elastic Bending Energy in Three Dimensions, J. Computational Phys., 212, no2, pp.757-777, 2006.
3-d simulations
Retrieving Topological Information for Phase Field Models, SIAM J. Appl. Math, 65, pp.1913-1932, 2005
How to effectively recover useful statistics such as topological information from phase field simulations without doing surface tracking? We provide an approach
Morphology of critical nuclei in solid state phase transformations, Physical Review Letters 98, 265703, 2007
using the minimax method and diffuse interface framework for critical nuclei computation
Diffuse-Interface Description of Strain-Dominated Morphology of Critical Nuclei in Phase Transformations, Acta Materialia, 56, pp.3568-3576, 2008.
Three dimensional simulations of critical nuclei morphology.
Mathematical and numerical aspects of phase-field approach to critical nuclei morphology in solids, J. Scientific Computing, 37, pp.89-102, 2008.
(special volume to the 2007 John Barrett Memorial Lectures), verifying Palais-Smale condition and spectral accuracy
A Variational Construction of Anistropic Mobility in Phase-Field Simulation , Discrete and Continous Dynamic Systems, 6, pp.391-406, 2006
Spectral Implementation of An Adaptive Moving Mesh Method for Phase-field Equations J. Computational Phys., 220, pp.498-510, 2006.
Analysis of a Phase Field Navier-Stokes Vesicle-Fluid Interaction Model, Disc. Cont. Dyn. Sys. B, 8, No3, pp.539-556, 2007.

Modeling material defect

An integrated Framework for multi-scale materials simulation and design, J. Computer Aided Material Design, 11, pp.183-199, 2004
Modeling and Computation of Random Thermal Fluctuations and Material Defects in the Ginzburg-Landau Model for Superconductivity ,
J. Computational Physics, 181, 45-67, 2002
An iterative-perturbation scheme for treating inhomogeneous elasticity in phase-field models ,
J. Computational Phys., 208, pp.34-50, 2005
Numerical approximations of the Ginzburg-Landau Models for Superconductivity, J. Mathematical Physics, 46, pp.095109(1-22), 2005
(an invited survey article)

Phase diagram

A new algorithm for the automation of phase diagram calculation, Computational Materials Sciences, 35, pp.61-74, 2006.

Liquid crystal

Fourier Spectral Approximation to a Dissipative System Modeling the Flow of Liquid Crystals, SIAM Numer. Anal. 39, pp. 735-762. 2001

Computational biology

(research supported in part by NSF DMS)

Shape of red blood cell:
Discocytes, resulting from simulation
using phase field models

Cell membrane

The usual vesicle membranes are formed by bilayers of lipid molecules. These lipid membranes exist everywhere in life and compartmentalize living matter into cells and subcellular structures and present themselves as highly structured interfaces which are essential for many biological functions.
The equilibrium shapes of bilayer vesicle membranes have been successfully modeled via the minimization of certain shape energy such as the elastic bending energy. Recently, we have developed effective phase field bending elasticity models and simulation tools for such problems.

A phase field approach in the numerical study of the elastic bending energy for vesicle membranes , J. Computational Physics, 198, pp450-468, 2004

Retrieving Topological Information for Phase Field Models, SIAM J. Appl. Math, 65, pp.1913-1932, 2005

A phase field formulation of the Willmore problem. Nonlinearity, 18, pp.1249-1267, 2005.

Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation , Comm Pure Appl Anal, 4, pp.537-548, 2005

Simulating the Deformation of Vesicle Membranes under Elastic Bending Energy in Three Dimensions, J. Computational Phys., 212, no2, pp.757-777, 2006.

Modeling Vesicle Deformations in Flow Fiel ds via Energetic Variational Approaches, submitted for publication, 2005

Analysis of a Mixed Finite Element Method for a Phase Field Bending Elasticity Model of Vesicle Membrane Deformation J. Computational Math., 2006.

Convergence of numerical approximations to a phase field bending elasticity model of membrane deformations, Inter. J. Numer. Anal and Modeling, 4, pp.441-459, 2006

Analysis of a Phase Field Navier-Stokes Vesicle-Fluid Interaction Model, Disc. Cont. Dyn. Sys. B, 8, No3, pp.539-556, 2007

Modelling and Simulations of Multi-component Lipid Membranes and Open Membranes via Diffuse Interface Approaches, J. Mathematical Biology, 56, no3, pp.347-371, 2008

Adaptive Finite Element Method for a Phase Field Bending Elasticity Model of Vesicle Membrane Deformations, 30, no3, pp.1634-1657, 2008

Adhesion of vesicles to curved substrates, Phys. Rev. E, 77, 011907(1-7), 2008

Cell aggregation/adhesion

We recently started collaborations with bioengineers to study the tumor metastasis. We focused on the population balance equation approach for the modeling and simulation of cell aggregation and adhesion.

Monte Carlo simulation of heterotypic cell aggregation in nonlinear shear flow Mathematical Biosciences and Engineering, 3, pp.683-696, 2006

Population balance equations with bounded kernels and their numerical approximation, preprint, 2006

Computational fluid

Impulsive Stretching of a Surface in a Viscous Fluid, SIAM Appl. Math., 57, pp. 1-14, 1997
FEM and control of Ladyzhenskaya models, Comp. Math.
Finite-Element Approximations of a Ladyzhenskaya Model for Stationary Incompressible Viscous Flow, SIAM J. Numer. Anal., 27, pp1-19, 1990.
Analysis of a fluid elastic structure interaction model, Dis. Cont. Dyn. Systems, 9, pp.633--650, 2003
Fourier Spectral Approximation to a Dissipative System Modeling the Flow of Liquid Crystals, SIAM J. Numer. Anal., 39, pp. 735-762, 2001.
Semidiscrete Finite Element Approximations of a Linear Fluid-Structure Interaction Problem, SIAM Num. Anal., 42, pp. 1-29, 2004
From Micro to Macro Dynamics via a New Closure Approximation to the FENE Model of Polymeric Fluids, SIAM Multiscale Model. Simul., 3, pp. 895-917, 2005
FENE Dumbbell Model and Its Several Linear and Nonlinear Closure Approximations , with P. Yu and C. Liu, SIAM Multiscale Model. Simul., 4, pp. 709-731, 2005
An enhanced macroscopic closure approximation to the micro-macro FENE models for polymeric materials, SIAM Multiscale Modelling and Simulations, 2008
A Maximum Entropy Principle Based Closure Method for Macro-Micro Models of Polymeric Materials, Kinetic and Re lated Models, 1, no2, pp.171-184, 2008

3D Meshing and Centroidal Voronoi Tessellation

(CVT generated tetrahedral mesh for a cube with 9 balls enclosed, more pictures may be found in the papers)

A centroidal Voronoi tessellation is a Voronoi tessellation of a given set such that the
associated generating points are centroids (centers of mass with respect to a given
density function) of the corresponding Voronoi regions. Such tessellations are useful,
in among many other contexts, data compression, optimal quadrature rules, optimal
representation and quantization, image analysis, finite difference and volume schemes,
mesh generations, optimal distribution of resources, cellular biology, and the territorial
behavior of animals. We are studying methods for computing these tessellations, the
underlying mathematical theory and their applications.

Centroidal Voronoi Tessellations: Applications and Algorithms, SIAM Review, 41, p637-676, 1999
discuss the basic concept of CVT and various applications, algorithms
Search Voronoi Tessellations on Google Scholar for relevant links

Constrained Centroidal Voronoi Tessellations for Surfaces, SIAM J. Scientific Comp., 24, pp. 1488-1506, 2003
discuss how to compute constrained CVTs, use projection!

Grid generation and optimization based on centroidal Voronoi tessellations , Appl Math Comp, 133, pp591-607, 2002
discuss how to use CVT to optimize (generate) grid

Meshfree, probabilistic determination of point sets and support regions for meshless computing , Computer Meth in Applied Mech Eng, 191, pp.1349-1366, 2002
how to get nodes in truly meshless fashion? we give an answer

Numerical studies of MacQueen's k-means algorithm for computing the centroidal voronoi tessellations , Computer Math Appl 44, 511-523, 2002

CVT based POD analysis, International series of Numerical Mathematics, 143, pp.137-150, Birkhauser, 2002

Voronoi-based finite volume methods, optimal Voronoi meshes, and PDEs on the sphere Comput. Meth. Appl. Mech. Engrg. , 192, pp.3933-3957, 2003;

Finite volume methods and spherical centroidal Voronoi tessellations preprint

Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations , Inter. J. Numer. Meth. Eng., 56, No.9, pp.1355-1373, 2002.
more examples of CVT basd tetrahrdral mesh

Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations , Parallel Computing, 28, pp.1477-1500, 2002
Our paper in Parallel Computing was in the top download at Elsevier in 2003.

Reduced Order Modeling of Complex Systems, Proceedings of the 20th. Dundee Numerical Analysis Conference, 2003

Anisotropic Centroidal Voronoi Tessellations and Their Applications, SIAM J. Sci Comp., 26, pp. 737-761, 2005

Boundary recovery for three dimensional conforming Delaunay triangulation, Comp. Meth. Appl. Mech. Engr

Constrained boundary recovery for three dimensional Delaunay triangulation, Int. J. Numer. Meth. Eng.

Tessellation and Clustering by Mixture Models and Their Parallel Implementations, SIAM Data Mining Conference, 2004

Recent progress in robust and quality Delaunay mesh generation , J. Computational Applied Mathematics, 195, pp.8-23, 2006.

Convergence of the Lloyd Algorithm for Computing Centroidal Voronoi Tessellations, SIAM J. Numer. Anal. , 44, pp. 102-119, 2006

Acceleration of algorithms for the computation of centroidal Voronoi tessellations, Numerical Linear Algebra and Applications, 13, pp.173 - 192, 2006

Centroidal Voronoi tessellation algorithms for image processing , J. of Mathematical Imaging and Vision, 24, pp.177-194, 2006

Ideal Point Distributions, Best Mode Selections and Optimal Spatial Partitions via Centroidal Voronoi Tessellations , Proceeding of 2nd Inter Symposium on Voronoi Diagrams in Sciences and Engineering, Seoul, Korea, Oct, 2005

papers availble upon request

Mathematical Modelling

Structural stability

Study of an inclined, segmented cantilever with tip load, Quart. Applied Math., 53, pp 81-93, 1995

Nonlinear hyperbolic conservation laws

Spectral viscosity methods for multidimensional conservation laws, Math. Comp. 61, 629-643, 1993
A low dispersion, high accuracy finite element method for first order hyperbolic systems in several space variables}, Comp. Math Appl., 15 447-457, 1988

Stochastic and multiscale modelling

Stochastic dynamics of Ginzburg-Landau vortices in superconductors Phys. Rev B, pp.52506, 2001

Numerical Approximation of Some Linear Stochastic Partial Differential Equations Driven by Special Additive Noises, SIAM J. Numer. Anal., 40, pp.1421-1445, 2002

Modeling and Computation of Random Thermal Fluctuations and Material Defects in the Ginzburg-Landau Model for Superconductivity , J. Computational Physics, 181, no.1, p.45-67, 2002

From Micro to Macro Dynamics via a New Closure Approximation to the FENE Model of Polymeric Fluids, --> SIAM Multiscale Model. Simul., 3, pp. 895-917, 2005

FENE Dumbbell Model and Its Several Linear and Nonlinear Closure Approximations , with P. Yu and C. Liu, SIAM Multiscale Model. Simul., 3, pp. 709-731, 2005

Parallel algorithms

Domain decomposition

A finite difference domain decomposition algorithm for numerical solution of the heat equation, Mathematics of Computation, 57, pp63-71, 1991

FEM domain decomposition for parabolic equations, Domain decomposition methods for PDEs (SIAM), pp255-263, 1991

Optimization Based Nonoverlapping Domain Decomposition Algorithms and Their Convergence, SIAM Numer. Anal., 39, pp. 1056-1077, 2001

A Gradient Method Approach to Optimization-Based Multidisciplinary Simulations and Nonoverlapping Domain Decomposition Algorithms, SIAM Numer. Anal. 37, 1513-1541, 2000

Efficient Parallel Algorithms for Parabolic Problems, SIAM Numer Anal, 39, pp. 1469-1487, 2002

Parallel implementaion

Numerical Algorithms of the Lawrence--Doniach Model for Layered Superconductors and their Parallel Implementation, SIAM J. Sci. Comp., 20, pp. 2122-2139, 1999

Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations , Parallel Computing, 28, pp.1477-1500, 2002

Divide and Conquer for eigenproblems

Quasi-Laguerre Iteration in Solving Symmetric Tridiagonal Eigenvalue Problems, SIAM J. Sci. Comp., 17, pp. 1347-1368, 1996

What's new in polynomial solving and eigen-problems?

Quasi-Laguerre Iteration in Solving Symmetric Tridiagonal Eigenvalue Problems, SIAM J. Sci. Comp., 17, pp. 1347-1368, 1996

The Quasi-Laguerre iteration, Math. Comp.

Quasi-Laguerre iterations, (lecture notes)

Others available upon request

Mathematical analysis, PDEs, Operators

Ginzburg-Landau Vortices: Dynamics, Pinning, and Hysteresis, SIAM Math. Anal., 28, pp. 1265-1293, 1997
Existence of solution for time-dependent G-L equations, Applicable Anal., 1994
Existence of Weak Solutions to Some Vortex Density Models, SIAM Math. Anal., 34, pp. 1279-1299, 2003
Critical Magnetic Field and Asymptotic Behavior of Superconducting Thin Films, SIAM Math. Anal., 34, pp. 239-256, 2002
The Global Minimizers and vortex solutions to a Ginzburg-Landau model of superconducting films, Comm Pure Appl Anal, 2002
Approximation of Fronbenius-Perron opertors Appl. Math. Comp., 53, pp151-171, 1993
Others available upon request

Contact Qiang Du 2003-04-08