CCMA Seminars on PDE & Numerical Methods Spring 2006 

DATE  SPEAKER  TITLE 
01/30/06  Changyou Wang
University of Kentucky 
TITLE: Stablestationary harmonic maps to spheres
Abstract: I will first discuss a sharp version of an improved Kato's inequality for smooth harmonic maps between manifolds. Then use it to obtain new estimates on the Hausdorff dimension of singular set for stablestationary harmonic maps to spheres. This is joint work with F.H. Lin. 
02/13/06  Xinwei Yu
UCLA 
Level Set Dynamics and the Nonblowup of the 2D Quasigeostrophic
Equation
Abstract: The 2D Surface Quasigeostrophic (SQG) equation describes the transportation of an "active scalar" by a divergence free velocity field. It draws much interest in the PDE community recently after being suggested by Constantin, Majda and Tabak as a nontrivial model for the study of possible finite time singularity formation for the 3D incompressible Euler equations. It turned out that whether there are finite time singularities in the 2D SQG flow is also a very challenging problem and remains open today. In this talk, I will present a novel approach for the study of the singularity problem for the SQG equation. It estimates the growth of the gradient of the active scalar from a pure Lagrangian point of view. Under certain assumptions, this new approach excludes the formation of singularities and furthermore yields sharp growth estimates. I will further present numerical simulations which suggest that the aforementioned assumptions are practical and therefore reasonable. I will also mention recent progresses of the theoretical and numerical study of the singularity problem for the 3D incompressible Euler equations. This is joint work done with Jian Deng, Thomas Y. Hou and Ruo Li. 
02/27/06  Ping Lin
National University of Singapore 
Title: An Energylaw Preserving Scheme and C^0 Finite Element Methods for Singularity Dynamics of Liquid Crystal Flows Anstract: Liquid crystal flow model is a coupling between orientation (director field) of liquid crystal molecules and a flow field. The model is also related to a phase field model of multiphase flows and to microfluidics device. It is crucial to preserve the energy laws of the hydrodynamical system in numerical simulation of liquid crystal flows, especially when orientation singularities are involved. We shall use a C^0 finite element method which is simpler than existing C^1 element methods and mixed element formulation. Through a reformulation the energy law can be achieved by the C^0 finite element method. A discrete energy law is achieved for an explicitimplicit time discretization and a special treatment of the nonlinear phase change term. Apparently the discrete energy law is an approximation of the continous energy law. A characteristic finite element method combined with a few fixed point iterations automatically separates the flow and director equations and thus reduces the size of the stiffness matrix and at the same time keeps the stiffness matrix time independent. The latter avoids solving a linear system at every time step and largely reduces the computational time, especially when direct linear system solvers are used. We will consider both smaller and larger liquid crystal molecule cases. A number of examples are computed to demonstrate the algorithm. 
03/6/06  Dionisios Margetis
MIT 
Mathematical modeling of crystal surfaces: From discrete schemes to continuum laws Dionisios Margetis Department of Mathematics, M.I.T. Abstract: In traditional settings such as fluids and classical elasticity the starting point (``truth'') is identified with continuum equations. But in many cases of mathematical modeling this perspective is changed: The truth is atomistic, or takes the form of discrete schemes, by which continuum laws must be determined at the macroscale. The issue of connections between different scales is largely unresolved. In this talk I focus on the evolution of crystal surfaces as a prototypical case of coupling between scales, with implications in the design of novel devices. The governing, discrete equations represent the motion of interacting line defects, atomic ``steps''. In the continuum limit a nonlinear PDE is derived for the surface height, and freeboundary problems are formulated for the surface motion. I show analytically how microscopic details of the crystal enter the requisite boundary conditions, and thus affect evolution at the macroscale. 
03/13/06  Hennie Poulisse
Shell 
TITLE: Best Approximation in an Inner Product Space from the Range of a Linear Operator over a Polyhedron with an Application to Oil Production SPEAKER: Hennie Poulisse ABSTRACT: The range of the linear operator over a polyhedron considered in this paper is a convex subset H of a finite dimensional subspace S of the ambient inner product space X. According to the reduction principle the best approximation to a point of X from H equals the best approximation to the best approximation to that point of X from S from H. Equivalent representations are given for H in terms of ^Ñopposite^Ò translated convex cones. Moreover translated polar cones are introduced. Subsequently, explicit calculations are given for the best approximations from a translated convex cone and from a translated polar cone. Using the BoyleDykstra theorem, these results are used to give a constructive proof of the best approximation to a point of X from H. The result is finally applied to reallife data from oil industry in that a solution is presented for a very important problem in oil  and gas production operations called the reconciliation problem, where the contribution of individual wells to a measured total production has to be assessed. 
03/27/06  Do Y. Kwak
Korea Adv. Institute of Sci. Tech. 
Title : Mixed finite element methods for general quadrilateral grids Abstract : In this talk, we show some peculiar aspects of mixed finite elements for general quadrilateral grids for elliptic problems. As with scalar finite element methods, all the functions are defined on a reference element and then mapped onto the general element via certain mapping. Scalar functions are mapped in the usual way, by composition. Meanwhile, vector functions are mapped by a special map, called Piola map to preserve divergence of the function. This does not cause much difficulties in case of triangular grids. However, some problems arise when quadrilateral grids are used: First, the divergence of approximate velocity space loses optimal approximation property due to the lack of proper polynomials: Second, the divergence of vector fields no longer lie in the pressure(scalar) field, which causes stability problem. Based on the above observation, we suggest some new element which is a modification of RaviartThomas element of lowest order. This new element is designed so that the divergence of velocity fields lie in the pressure space, and $H(div)$projection $\Pi_h$ satisfies $div\cdot\Pi_h=P_h div$. A rigorous optimal order error estimate is carried out by proving a modified version of the BrambleHilbert lemma for vector variables. We show a local $H(div)$projection reproducing certain polynomials suffices to yield an optimal $L^2$error estimate for the velocity and hence our approach also provides an improved error estimate for original RaviartThomas element of lowest order. 
03/30/06 (4:00pm, rm 106)  Helge Holden
NTNU/NUST 
On convergent difference schemes for the HunterSaxton equation 
04/03/06  Tong Li
University of Iowa 
Transition to Instabilities of Discrete Shocks 
04/10/06  Fioralba Cakoni
University of Delaware 
Title: Mixed boundary value problems in inverse electromagnetic scattering theory. We consider the inverse scattering problem of determining the shape and physical properties of an obstacle from a knowledge of the scattered field due to the scattering of an incident timeharmonic electromagnetic wave at fixed frequency. In particular, we discuss the linear sampling method for solving the inverse problem in the case of mixed boundary values problems for Maxwell equations. Mixed boundary value problems in electromagnetic scattering theory arise when the scattering object is a composite material such that parts of the scatterer have different electrical properties. Such objects can be partially coated perfect conductors or dielectrics, thin objects with one side a perfect conductor and the other side an imperfect conductor or dielectric. The mathematical analysis of the direct problem is difficult due to the nonstandard solution space. Furthermore, no matter how smooth the boundary data is, the change of boundary conditions causes the scattered field to be singular at the interface, which gives rise to numerical difficulties. Concerning the corresponding inverse problem, since the physical structure of the composite medium is not known a priori, the use of weak scattering approximations and/or nonlinear optimization techniques are problematic. We will also discuss some related open problems and present examples of reconstruction showing the viability of our method for solving the inverse scattering problem. 
04/12/06(3:30 in Room 216)  Zhimin Zhang
Wayne State University 
Title: High accuracy eigenvalue approximations Abstract: Finite element methods, spectral methods, and spectral collocation methods are applied to approximate some eigenvalue problems based on the nature of the problem. Newly developed gradient recovery techniques are adopted to enhance the accuracy of the eigenvalue approximation. 
04/17/06 (3:30pm, Room 216)  Anja Schloemerkemper
MPI, Leipzig 
Analysis and numerical simulation of magnetic forces between rigid polygonal bodies Abstract: We discuss magnetic force formulae for rigid magnetized bodies. To derive such formulae for bodies in contact, we work on the one hand in the context of macroscopic electrodynamics, on the other hand we start with a discrete setting of magnetic dipole moments and derive the continuum limit. After I will have delivered an insight into the mathematical proofs, I will present some numerical experiments. These study the dependence of the magnetic force on the distance between magnetic bodies and allow to compare the force formulae for bodies being in contact quantitatively. Finally, I will discuss the results from a physical pointofview. 
05/08/06(10:00am in Room 106)  Peter L. Shi
Oakland University 
Title: How to solve nonlinear finite element models with optimal speed? Abstract: The speaker will present a new method for solving nonlinear finite element models in optimal speed. The method, in a unified framework, will handle boundary value problems of elliptic systems whose principle part is Lipschitz continuous and strongly monotone in a Sobolev space and whose lower order terms are subject to certain growth conditions. Numerical examples will also be presented. 
05/08/06  Nick Costanzino
UNC 
Title: Traveling waves of the Ostrovsky equation and generalizations. Abstract: The Ostrovsky equation is a model for internal surface waves of the ocean which includes the Coriolis force. After a suitable rescaling, the Ostrovsky equation is similar to the regularized short pulse equation, which is a recently derived model for the propagation of short optical pulses in a fiber. We discuss the various aspects of traveling waves associated with these equations, including the existence and nonexistence of single bump traveling waves, the existence of an orbitflip bifurcation, and the existence of multiple bump waves. The stability of these waves will also be discussed. This is joint work with Christopher Jones and Vahagn Manukian (UNCChapel Hill). 


Sponsored in part by CCMA and by individual faculty grants  
