• August 29, 2014
    Junshan Lin
    Department of Mathematics and Statistics, Auburn University
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Electromagnetic Field Enhancement for Metallic Nano-gaps

    Abstract: There has been increasing interests in electromagnetic field enhancement and extraordinary optical transmission effect through subwavelength apertures in recent years, due to its significant potential applications in biological and chemical sensing, spectroscopy, terahertz semiconductor devices, etc. In this talk, I will present a quantitative analysis for the field enhancement when an electromagnetic wave passes through small metallic gaps. We focus on a particular configuration when there is extreme scale difference between the wavelength of the incident wave, the thickness of metal films, and the size of gap aperture. Based upon a rigorous study of the perfect electrical conductor model, we show that enormous electric field enhancement occurs inside the gap. Furthermore, the enhancement strength is proportional to ratio between the wavelength of the incident wave and the thickness of the metal film, which could exceed 10000 due to the scale difference between the two. On other hand, there is no significant magnetic field enhancement inside the gap. The ongoing work along this research direction will also be discussed.

  • September 12, 2014
    Shan Zhao
    Department of Mathematics, University of Alabama
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: New Developments of Alternating Direction Implicit (ADI) Algorithms for Biomolecular Solvation Analysis

    Abstract: In this talk, I will first present some tailored alternating direction implicit (ADI) algorithms for solving nonlinear PDEs in biomolecular solvation analysis. Based on the variational formulation, we have previously proposed a pseudo-transient continuation model to couple a nonlinear Poisson–Boltzmann (NPB) equation for the electrostatic potential with a geometric flow equation defining the biomolecular surface. To speed up the simulation, we have reformulated the geometric flow equation so that an unconditionally stable ADI algorithm can be realized for molecular surface generation. Meanwhile, to overcome the stability issue associated with the strong nonlinearity, we have introduced an operator splitting ADI method for solving the NPB equation. Motivated by our biological applications, we have also recently carried out some studies on the algorithm development for solving the parabolic interface problem. A novel matched ADI method has been developed to solve a 2D diffusion equation with material interfaces involving complex geometries. For the first time in the literature, the ADI finite difference method is able to deliver a second order of accuracy in space for arbitrarily shaped interfaces and spatial-temporal dependent interface conditions.

  • September 19, 2014
    Catalin Turc
    Department of Mathematics and Statistics, New Jersey Institute of Technology
    Location and Time: Parker Hall 328, 2pm-3pm
    Title:Well-conditioned Boundary Integral Equation Formulations for the Solution of High-frequency Scattering Problems

    Abstract: We present several versions of Regularized Combined Field Integral Equation (CFIER) formulations for the solution of two and three dimensional frequency domain scattering problems with various kinds of boundary conditions. These formulations are based on suitable approximations to Dirichlet-to-Neuman operators and can be shown to be well posed , under certain assumptions on the regularity of the scatterers. For a wide variety of scatterers, solvers based on these formulations outperform solvers based on the classical Combined Field Integral Equations.

  • September 26, 2014
    Jiayin Jin
    Department of Mathematics, Michigan Sate University
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Global Dynamics of Boundary Droplets for the 2-d Mass-conserving Allen-Cahn Equation

    Abstract: In this talk I will present how to establish the existence of a invariant manifold of bubble states for the mass-conserving Allen-Cahn equation in two space dimensions, and give the dynamics for the center of the bubble.

  • October 3, 2014
    Zhongwei Shen
    Department of Mathematics and Statistics, Auburn University
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Front Propagation in Reaction-Diffusion Equations with Ignition Nonlinearities

    Abstract:In this talk, I will present the developments of front propagation in diffusive media of ignition type, with the focus on traveling waves and their generalizations. I will first present some classical results in the homogeneous media. Then, I will move to the recent developments in space heterogeneous media. Finally, I will present my recent work with W. Shen in time heterogeneous media.

  • October 10, 2014
    Anthony Skjellum
    Department of Computer Science and Software Engineering, Auburn University
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Toward Fault Tolerant Parallel Computing with MPI

    Abstract: In this talk, I present issues with the MPI parallel programming model, and the emerging issues of fault tolerance at scale. Over the past fifteen years, various efforts have been made to make MPI and parallel programs in general more tolerant to faults. We describe two current models - FA-MPI and ULFM, which are current models for making MPI fault tolerant, or at least enabling fault tolerance with MPI applications. We also describe basic reasons for faults, how emerging architectures are leading to higher fault rates, and where other opportunities coming from mathematical analysis and algorithm theory can help address problems in such scenarios such as Newton-Krylov solvers inside simulation applications.

  • October 24, 2014
    Wenxian Shen (canceled)
    Department of Mathematics and Statistics, Auburn University
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: TBA

    Abstract: TBA

  • October 31, 2014 (special applied math seminar)
    Jie Shen
    Department of Mathematics, Purdue University
    Location and Time: Parker Hall 250, 4pm-5pm
    Title:Phase-field Models for Multiphase Complex Fluids: Modeling, Numerical analysis and Simulations

    Abstract: I shall present some recent work on phase-field model for multiphase incompressible flows. We shall pay particular attention to situations with large density ratios as they lead to formidable challenges in both analysis and simulation. I shall present efficient and accurate numerical schemes for solving this coupled nonlinear system, in many case prove that they are energy stable, and show ample numerical results which not only demonstrate the effectiveness of the numerical schemes, but also validate the flexibility and robustness of the phase-field model.

  • November 7, 2014
    Richard Zalik
    Department of Mathematics and Statistics, Auburn University
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: On the Nonlinear Jeffcott Equations

    Abstract:The Jeffcott equations are a system of coupled, nonlinear, ordinary differential equations. The primary application of their study is directed towards understanding the reasons for the excessive vibrations recorded in the cryogenic pumps of the second stage main engine of the Space Shuttle, during hot firing ground testing. In this talk we shall examine some properties of the solutions of the Jeffcott equations. In particular, we show how bounds for the solutions of these equations can be obtained from bounds of the solutions of the linearized equations. By studying the behavior of the Fourier transforms of the solutions, we are also able to predict the onset of destructive vibrations. These conclusions are verified by means of numerical solutions of the equations, and of power spectrum density plots. This work shows how numerical simulations can be used to obtain an insight into the correct solution to a problem. Once this correct answer is known, it then becomes possible to give a rigorous proof.

  • November 14, 2014
    Michael Neilan
    Department of Mathematics, University of Pittsburgh
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Finite Element Methods for Elliptic Problems in Non-divergence Form

    Abstract:The finite element method is a powerful and ubiquitous tool in numerical analysis and scientific computing to compute approximate solutions to partial differential equations (PDEs). A contributing factor of the method's success is that it naturally fits into the functional analysis framework of variational models. In this talk I will discuss finite element methods for PDEs problems that do not conform to the usual variational framework, namely, elliptic PDEs in non--divergence form. I will first present the derivation of the scheme and give a brief outline of the convergence analysis. Finally, I will present several challenging numerical examples showing the robustness of the method as well as verifying the theoretical results.