CCMA Seminars on PDE & Numerical Methods- Spring 2007

Location and time: Penn State University
Department of Mathematics
3:30pm --- 4:30pm
Monday (even weeks)
216 McAllister Building

01/09/2007 Jichun Li
University of Nevada
(Room 106)
Title: Finite Element Analysis for Time-Dependent Maxwell's Equations in Complex Media

Abstract: In this talk, we will discuss time-dependent Maxwell^s equations in dispersive media on a bounded three-dimensional domain. Both standard and mixed finite element methods are developed and error estimates are proved for both semi- and fully-discrete schemes. Three most popular dispersive media models (cold plasma, one-pole Debye medium and two-pole Lorentz medium) and double-negative metamaterial model will be discussed. Some preliminary numerical results will be presented also.
02/05/07 Wei Zhu
Rice University
TITLE: An energy reducing flow for multiple-valued functions.

abstract: By the method of discrete Morse flows, we construct an energy reducing multiple valued function flow. The flow we get is Holder continuous with respect to the L2 norm. We also give another way of constructing flows in some special cases, where the flow we get behaves like the ordinary heat flow.
02/12/07 Giuseppe Coclite
University of Bari
TITLE and Abstract

02/19/07 Guang Qin
Texas A&M
TITLE and Abstract

02/26/07 Pilhwa Lee
Courant Institute
Title:Adaptive and Implicit Immersed Boundary Method with Advection- Electrodiffusion Abstract: Immersed boundary method is a mathematical and computational framework involving the interaction of fluid and structure. Advection- electrodiffusion of ions(solutes) dissolved in fluid in a biological system is considered in the context of fluid-solute-structure. For a chemical barrier across a boundary, a chemical potential is placed for each solute(ion) along the boundary. An implicit numerical scheme is proposed. For numerical accuracy and efficiency, adaptive mesh refinement around boundaries is applied. The Stokes equations are solved with hybrid approximate projection method with cell-centered grid. Advection-electrodiffusion equations are solved with the combination of geometric and algebraic multigrid methods. With advection of the boundaries, advection of ions is observed. With local change of the amplitude of chemical potentials, diffusion of each ion across the boundaries is regulated. The results show electroneutrality except in space charge layers near membranes, and agrees with the Nernst equation for the potential difference across membranes.

03/05/07 Carme Calderer
University of Minnesota
Mathematical modeling of elastomers and gels

Abstract: Since the celebrated existence theorems in nonlinear elasticity by John Ball in 1977, many applications emerged in the study of complex material phenomena involving diffusiveless phase transitions. This talk examines recent applications of caluclus of variations to study liquid crystal elastomers and gels. In the former, we explore the significantly different properties of the energy that result from combining the anisotropy of liquid crystals with the nonlinear elastic properties of polymers. In studies of gels, we exmaine how the combination of chemistry and network elasticity opens up to exciting new applications in the field of pharmacology. The work reported is join with Chun Liu, Baisheng Yan, Aaron Yip, Hang Zhang and Ronald Seigel.
03/19/07 Jorge Sofo

03/26/07 Robert Falgout
Compatible Relaxation in Algebraic Multigrid Methods

Abstract: Algebraic multigrid (AMG) is an important method for solving the large sparse linear systems that arise in many PDE-based scientific simulation codes. A major component of algebraic multigrid methods is the selection of coarse grids and the construction of interpolation. The notion of {\em compatible relaxation} (CR) was introduced by Brandt in \cite{ABrandt_2000} as a modified relaxation scheme that keeps the coarse-level variables invariant. Brandt states that the convergence rate of CR is a general measure for the quality of the set of coarse variables, and in \cite{GAMG_2004}, we developed a supporting theory for this idea. We have since developed an algebraic coarsening algorithm based on compatible relaxation that has several nice properties over the classical coarsening schemes. One such characteristic is its ability to ensure the quality of the coarse grid. In this talk, we will review the theory behind CR, describe our CR coarsening algorithm, and discuss aspects of the method that require additional development such as coarsening for systems of PDEs. We will also discuss CR's ability to predict the convergence behavior of the AMG method, and its potential role in adaptive AMG methods. Finally, we will talk about issues of parallelizing these methods to run on massively parallel computers. This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. \begin{thebibliography}{1} \small \bibitem{ABrandt_2000} {\sc A.~Brandt}, {\em General highly accurate algebraic coarsening}, Electronic Transactions on Numerical Analysis, 10 (2000), pp.~1--20. \bibitem{GAMG_2004} {\sc R.~D. Falgout and P.~S. Vassilevski}, {\em On generalizing the {AMG} framework}, {SIAM} J. Numer. Anal., 42 (2004), pp.~1669--1693. UCRL-JC-150807. \end{thebibliography}
04/02/07 (11:00 am) Ismael Herrera Revilla

Abstract: 1;5B
04/02/07 Alejandro Rey
McGill University

Abstract: Liquid crystal phases are found in DNA, proteins, lipids and polysaccharides. Frozen-in, chiral liquid crystal ordering also occurs in solid biocomposites such as insect cuticle, muscle, plant cell walls and collagen, where the helicoid structure is believed to arise by self-assembly processes. Spinning of silk fibers by spiders is another biological polymer process that relies on liquid crystal self-assembly. I will discuss the progress and challenges of modeling in three such applications: (1) Biological helicoids form by directed self-assembly. Theory and computer simulation of chiral phase ordering show that the directed self-assembly process reproduces the natural structures. The computational results shed light on the role of chiral ordering on the formation of helicoidal monodomains. (2) Spinning of spider silk involves a complex sequence of phase transitions that includes nematic phase ordering in the duct section of the spinning apparatus. Simulation of phase ordering under capillary confinement replicates the observed structures found in Nephila clavipes and other orb-weavers. The computational results shed light on the role of defect textures in the ber spinning process. (3) Biological membranes are smectic liquid crystals that display fexoelectricity, or coupling between electric fields and curvature. Models based on smectic elasticity and polarization thermodynamics are used to derive the electroelastic shape equation, whose solution gives the membrane shape under external fields. The theoretical results shed light on the various ways electric fields affect membrane shape and functioning.
04/09/07 Tiezheng Qian
A variational approach to moving contact line hydrodynamics

Abstract: In immiscible two-phase flows, contact line denotes the intersection of the fluid-fluid interface with the solid wall. When one fluid displaces the other, the contact line moves along the wall. A classical problem in continuum hydrodynamics is the incompatibility between the moving contact line and the no-slip boundary condition, as the latter leads to a non-integrable singularity. The recently discovered generalized Navier boundary condition (GNBC) offers an alternative to the no-slip boundary condition which can resolve the moving contact line conundrum. We present a variational derivation of the GNBC through the principle of minimum energy dissipation (entropy production), as formulated by Onsager for small perturbations away from the equilibrium. Through numerical implementation of a continuum hydrodynamic model, it is demonstrated that the GNBC can quantitatively reproduce the moving contact line slip velocity profiles obtained from molecular dynamics simulations. In particular, the transition from complete slip at the moving contact line to near-zero slip far away is shown to be governed by a power-law partial slip regime, extending to mesoscopic length scales.
04/16/07 Jesse Barlow
Penn State
Title: Some results in Gram--Schmidt and questions about block Gram--Schmidt

Abstract: Although classical Gram-Schmidt (CGS) has been with us for quite some time and is a standard topic in undergraduate linear algebra courses, there is still more to learn about it as an algorithm. Its main applications of interest are in the implementation of Krylov space methods such as GMRES or Arnoldi and in matrix modification problems. The difficulty is that the CGS taught in linear algebra classes has disasterous numerical properties unless reorthogonalization is done. Modified Gram--Schmidt (MGS) overcomes some of the numerical difficulties, but is slow on modern computer architectures; much slower than CGS with reorthogonalization. We explore some of the new results on CGS and we also look at a block version of CGS whose implementation will be necessary to obtain good efficiency. The results indicate that reorthogonalized CGS and reorthogonalized block CGS are likely to be part of GMRES/Arnoldi implementations in the future.
04/23/07 V. Rybalko
Variational Problem for the Ginzburg-Landau Functional with Degree Boundary Conditions

Abstract: In this talk we present recent results on existence/nonexistence of minimizers of Ginzburg-Landau functional with degree boundary condition. We will discuss how the rise of vortices is determined by the capacity of the domain and by the topology of the energy level sets.
05/07/07 Thomas Krainer
Penn State Altoona
Title: Elliptic boundary problems on a class of noncompact manifolds

Abstract: We discuss Fredholm criteria and regularity results for elliptic boundary value problems on a particular class of noncompact manifolds. An example for the topological setup would be Euclidean space with a noncompact obstacle removed, a model problem is the Schroedinger operator with (complex) potential and boundary conditions on the boundary of the obstacle. In general, the operators under consideration may analytically be regarded as cusp operators on manifolds with corners after suitable compactification of the noncompact ends, and boundary conditions are imposed on some of the boundary hypersurfaces. Cusp operators (with cusp degeneracy on the entire boundary) were introduced by Richard Melrose and Victor Nistor in 1996 in the context of the index problem on manifolds with corners of codimension 1 (unpublished), and in the case of higher codimensions by Robert Lauter and Sergiu Moroianu (2002).

Previous schedules: Fall 2006, Spring 2006, Fall 2005, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002,
Fall 2001, Spring 2001, Fall 2000, Fall 1999, Spring 1999, Spring 1998, Fall 1998.

For more information or to suggest speakers, please contact Chun Liu

Sponsored in part by CCMA and by individual faculty grants

Last modified: Tue 09/01/2005