Computational Methods and Applied PDE Workshop Schedule

Max Gunzburger  (Florida State University)
Least-squares Finite Element Methods for the Poisson Equation and  Their Connection to the Dirichlet and Kelvin Principles

Abstract:  Least-squares finite element methods for first-order formulations of  the Poisson equation are not subject to the inf-sup condition and  lead to stable solutions even when all variables are approximated by  equal-order continuous finite element spaces. For such elements, one  can also prove optimal convergence in the ``energy'' norm for all  variables and optimal L^2 convergence for the scalar variable.  However, showing optimal L^2 convergence for the flux has proven to  be impossible without adding the redundant curl equation to the first- order system of partial differential equations. In fact, numerical  evidence strongly suggests that nodal continuous flux approximations do not posses optimal L^2 accuracy. In this talk, we show that  optimal L^2 error rates for the flux can be achieved without the curl  constraint, provided that one uses the div-conforming family of  Brezzi-Douglas-Marini or Brezzi-Douglas-Duran-Fortin elements. Then,  we proceed to reveal an interesting connection between a least-squares finite element method involving div-conforming flux approximations and mixed finite element methods based on the classical Dirichlet and Kelvin principles. We show that such least- squares finite element methods can be obtained by approximating,  through an L^2 projection, the Hodge operator that connects the Kelvin and Dirichlet principles. Our principal conclusion is that  when implemented in this way, a least-squares finite element method  combines the best computational properties of mixed-Galerkin finite element methods based on each of the classical principles.

Chi-Wang Shu (Brown University )
Discontinuous Galerkin  Method Based on Non-polynomial Approximation Spaces

Abstract: We describe our recent development of discontinuous Galerkin (DG) methods based on non-polynomial approximation spaces for numerically solving time dependent hyperbolic and parabolic and steady state elliptic PDEs. The algorithm is based on approximation spaces consisting of non-polynomial elementary functions such as exponential functions, trigonometric functions, etc., with the objective of obtaining better approximations for specific types of PDEs and initial and boundary conditions. It is shown that L^2 stability and error estimates can be obtained when the approximation space is suitably selected. It is also shown with ample numerical examples that a careful selection of the approximation space to fit individual PDE and initial and boundary conditions often provide more accurate results than the DG methods based on polynomial approximation spaces of the same order of accuracy. This is joint work with Ling Yuan.

Marshall Slemrod (University of Wisconsin at Madison)  
Recent results on Dynamics of the Plasma Sheath

Abstract: In this lecture I will give an outline of the recent results on plasma sheath dynamics that were presented in papers with M Feldman and S Y Ha. The main contribution was to view that the dynamics of the sheath boundary layer in a plasma as an evolving level set where the accelaration ( not velocity) is proportional to the mean curvature.

Irene Gamba (University of Texas at Austin)
Deterministic solvers to Boltzmann-Poisson Dynamics

Abstract:  While the Boltzmann-Poisson system is the most reliable model for the flow of charged particles general practice has been to rely on stochastic solvers as Monte-Carlo (DSMC) methods. These solvers are computationally expensive for good resolution and unreliable for transient simulations.

We focus in a rather easy and fast deterministic high order solver to the self-consistent Boltzmann-Poisson system, where the collision operator incorporates acoustical and non-polar optical interactions such a phonon absorption and emission rates. We have modeled a short based channel flow in a solid semiconductor bounded structure, such as real MOSFET device models and bench-marked our calculations with DSMC solvers.

In particular, we shall present one and two space-dimension with three phase-velocity -dimension system of equations. They consists of a linear kinetic (non-local) equation solved by WENO methods coupled with the Poisson equation for the force field acting on the particles accounting for long range interactions.

We will focus on the development of the method, simulation results for diodes and MESFET and MOSFET as well as comparisons to other classical models in the field. In particular we compute, deterministically, the evolution probability density function with its first three moments.

Boundary singularities for 2-space dimensions models are described and accurately computed. This work has been done in collaboration with J.A. Carrillo, A. Majorana and C.-W. Shu.
 

Xiaoqiang Wang (The Institute of Mathematics and Applications (IMA))
Phase Field Models and Simulations of Vesicle Bio-Membranes

Abstract: Recently, we began to systematically model and simulate the shape deformation of vesicle membranes using a unified energetic variational phase field method based on the minimization of elastic bending energy with volume and surface area constraints. Mathematical theory and numerical algorithms are developed to for the phase field models. Rigorous convergence theories of the numerical methods are investigated. Many simulations are carried out in static and dynamic, axis-symmetric and full 3D, one component and multi-component cases. The new phase field modeling approach has the advantage of avoiding tracking the free interfaces, and it can easily handle topological changes. Meanwhile, a series of formulae for retrieving the Euler number of the vesicles have been given and discussed which may be useful for detection and control purposes. 

The 3D codes developed for the equilibrium shape deformations and the deformations and interactions with fluid fields allow us to conduct extensive computational studies. Both known and new equilibrium configurations have been discovered in our numerical simulations. A detailed examination of the energetic bifurcation landscape has been carried out. We have further studied the effect of the spontaneous curvature and have conducted simulations of vesicle transformations in fluids. The further development of the phase field approach for multicomponent vesicles gives us more tools to understand new and complex phenomena recently being experimentally studied by biologists.

Saturday, November 5, 2005

Tao Tang  ( Hong Kong Baptist University)
Gradient Stability and Large Time Stepping Methods for Nonlinear Diffusion Equations

Abstract:  We shall study numerical methods for nonlinear diffusion equations that describe complicated phase separation and coarsening phenomena. It is observed that most of the existing continuum model simulations have used explicit integration method in time and finite difference type approximation in space. In this case, the number of spatial grid points must be large and the time step has to be small in order to maintain the stability and to achieve high approximation accuracy. Even with rapidly increasing computational resources, explicit schemes are still limited to simulate early stage evolution or small length scale. It is therefore of practical importance to design more efficient simulation schemes.

The main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time-step than that for the usual semi-implicit scheme. Several numerical methods, including semi-implicit approaches and operator splitting methods will be investigated. Theoretical analysis on nonlinear stability will be also studied. Some rigorous mathematical theory for the underlying numerical schemes will be established. This talk will concentrate on two classes of nonlinear diffusion equations, namely the Cahn-Hilliard equation and the diffusion equations that model epitaxial growth of thin films.
 

Gregory Forest (University of North Carolina at Chapel Hill )
A Suite of Models and Algorithms for Nano-Composite Materials

Abstract:  The design of nano-composite materials is hindered by two significant bottlenecks:  physically accurate models (i.e., theory), and algorithms for the current and future models.  I will explain first why nano-composites present new property features beyond traditional composites, and the diversity of property targets.  Then, I will focus on current understanding of each phase of the nano-composite pipeline, from design components to final performance characterization.  The need for better & new theory and algorithms will become evident.

Poster  Presentations