Seminar meets Fridays at 2:00 pm in Parker 328 unless otherwise noted.

2014 - 2015

September 26

Speaker: **Jiayin Jin**, Department of Mathematics, Michigan Sate University

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.

September 19

Speaker: **Catalin Turc**, Department of Mathematical Sciences, New Jersey Institute of Technology

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 12

Speaker: **Shan Zhao**, Department of Mathematics, University of Alabama

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.

August 29

Speaker: **Junshan Lin**

Title: Electromagnetic Field Enhancement for Metallic Nano-gaps

Abstract: There has been increasing interest 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.