Abstract: In this talk, I review the mathematical results of the dynamcis of Bose-Einstein condensate (BEC) and present some efficient and stable numerical methods to compute ground states and dynamics of BEC. As preparatory steps, we take the 3D Gross-Pitaevskii equation (GPE) with an angular momentum rotation, scale it to obtain a four-parameter model and show how to reduce it to 2D GPE in certain limiting regimes. Then we study numerically and asymptotically the ground states, excited states and quantized vortex states as well as their energy and chemical potential diagram in rotating BEC. Some very interesting numerical results are observed. Finally, we study numerically stability and interaction of quantized vortices in rotating BEC. Some interesting interaction patterns will be reported.

Abstract: Research in public policy and public administration employ a variety of methodological and statistical approaches. Survey research, secondary data analysis, and field experiments are in use today. Often policy questions are studied over time and\or across states and communities. Researchers are becoming increasing aware of the problem that one unit of analysis, an individual or a community, may be nested within another unit. The presentation will cover topics under investigation by faculty and PhD students at UNLV and the potential for collaboration.

Abstract: I will attempt to combine an overview of our experience with hp finite elements and the automatic hp-adaptivity for wave propagation problems, with a presentation of new research topics focusing on multiphysics coupled problems. The first part of the talk will focus on fundamentals of hp-discretizations of wave propagation problems: acoustics, elasticity and electromagnetics. We will shortly discuss the issues of stability and approximability for time-harmonic problems emphasizing the difference between the elliptic and Maxwell problems. I will review the main results of the theory of projection-based interpolation and discuss its importance in both the theoretical (proof of discrete compactness for hp methods) and practical (automatic hp-adaptivity) context. This part of the presentation will deliver "punch lines" only, and I will finish it by "flashing" a few representative examples. The second part of the presentation will address our current work on multiphysics coupled-problems. Using a coupled acoustics/elasticity problem, I will outline new challenges that we have faced when generalizing the hp methodology to this class of problems. This will include a discussion on hp data structures, use of fractional Sobolev norms, and both energy- and goal-driven automatic hp-adaptivity algorithms. The discussion will be illustrated with numerical solutions of 3D axisymmetric problems.

Abstract: Solving Maxwell's equations plays an important role in many science and engineering areas. Examples include device design (such as antennas, radar and waveguides), nondestructive testing and imaging (geophysical probing for oil reservoirs and tumor detection), near field control and manipulation (detecting low levels of chemical and biological agents). In this talk I will start with three most popular dispersive media models: cold plasma, one-pole Debye medium and two-pole Lorentz medium. Next, I will show some finite element methods and corresponding error estimates for solving those models. Then I will extend the discussion to double negative metamaterials and present some numerical results. I will conclude the talk by posing some open problems and potential applications in nanotechnology and biomedical applications.

Abstract: Many phenomena in control schemes, biology, economics, engineering systems, physics, and other areas can be modeled by partial differential equations with stochastic perturbation terms. The numerical solution of Stochastic Partial Differential Equations (SPDEs) typically relies on (i) simulation or (ii) obtaining a semi-discretization in space and an application of time discretization methods to the resulting initial-value problem with Ito diffusions; both approaches generate a family of sample solutions (realizations) which must be averaged in order to determine the properties of the system. We introduce a hierarchical approximation of Fourier expansions which allows for a deterministic representation of the system’s aggregate properties. Applications from mathematical finance are considered in order to illustrate the approach.

Abstract: Finite element approximations for the eigenvalue roblem of the Laplace operator is discussed. A gradient recovery scheme is proposed to enhance the finite element solutions of the eigenvalues. By reconstructing the numerical gradient, it is possible to produce more accurate numerical eigenvalues. Furthermore, the recovered gradient can be used to form an {\it a posteriori} error estimator to guide an adaptive mesh refinement. Therefore, this method works not only for structured meshes, but also for unstructured and adaptive meshes. Additional computational cost for this post-processing technique is only $O(N)$ ($N$ is the total degrees of freedom), comparing with $O(N^2)$ cost for the original problem. Theoretical results can be summarized in the following: 1) Eigenfunctions are sufficiently smooth. Under uniform meshes or the Delaunay triangulation with regular refinement, the enhanced eigenvalue approximations for the linear element converge at rate $O(h^4)$ ($h$ is the maximum mesh size), while the original approximations converge at rate $O(h^2)$. 2) Eigenfunctions have corner singularities. Under a commonly used adaptive mesh refinement strategy, the enhanced eigenvalue approximations converge at rate $O(N^{-k/2-\rho})$ for some $\rho>0$ ($k=1$ for the linear element and $k=2$ for the quadratic element), while the original approximations converges at rate $O(N^{-k/2})$. All above theoretical results are numerically verified.

Abstract: Using computer simulations, we are now able to study the dependence of computed solutions on variations or uncertainties in the initial data, the forcing terms, or even in the coefficients or the physical properties of the system. The results of such studies suggest that both natural and engineering phenomena commonly framed in terms of deterministic systems of partial differential equations may be more correctly modeled and deeply understood as stochastic partial differential equations (SPDE's) instead. Stochastic models are more complex than deterministic ones; as part of this complexity, the solution of an SPDE is not simply a function, but rather a stochastic process which expresses the implicit variability of the system. This is the reason that SPDE's are able to more fully capture the behavior of interesting phenomena; it also means that the corresponding numerical analysis of the model will require new tools to model the systems, produce the solutions, and analyze the information stored within the solutions. In this talk, we will discuss some recent developments on numerical solutions for stochastic partial differential equations. In particular, we will present finite element approximations for a class nonlinear stochastic elliptic equations with white noise forcing terms and efficient Monte Carlo simulations for stochastic Helmholtz equations with random wavenumbers. Numerical experiments and simulations will also be illustrated.