Computational Mathematics Seminar

  • Thursday, December 8, 2011
    12:00 PM, Thackeray 427
     John Burkardt, Florida State University
    "Finite Element Methods for the (Navier) Stokes Equations "
     Abstract:
    This is not a research talk, but rather an introduction to some of the ideas, problems, techniques, algorithms, and programs associated with the application of the finite element method to flow equations. If you've set up the finite element equations for the Poisson problem in a rectangle, and wonder if the Navier Stokes equations will be just as easy, you are my ideal audience! Without spending too much time on mathematical or computational detail, I will try to present the Navier Stokes equations, simplify them as much as possible, and write down a weak form of the equations suitable for finite elements. Then I'll look at the construction of linear and quadratic basis functions, and the assembly of the system matrix. Finally, I'll discuss IFISS, a friendly, free and powerful PDE solver that works with MATLAB or OCTAVE. It includes a variety of elements and solvers, displays the solution graphically and computes error estimates.

  • Tuesday, November 15, 2011
    3:00 PM, Thackeray 704
     John Chrispell, Indiana University of Pennsylvania
    "Applications of Peskin's Immersed Boundary Method in Viscoelastic Fluids"
     Abstract:
    Modeling viscoelastic fluids driven by actuated immersed boundaries is of significance in both biological and industrial settings. To model these flows a nonlinear constitutive equation describing the evolution of the viscoelastic contribution to the fluid stress tensor is included in the governing equations. Here we discuss the use of an immersed boundary framework to simulate fluid flows governed by a Navier-Stokes/Oldroyd-B model. A description of the numerical method and its stabilization is given. We discuss recent numerical simulations involving the freely decaying shape oscillations of an Oldroyd-B fluid droplet suspended in an Oldroyd-B matrix, the peristaltic pumping of solid particles, as well as locomotion and hydrodynamic synchronization of undulating sheets in viscoelastic fluids.

  • Tuesday, November 1, 2011
    3:00 PM, Thackeray 704
     William C. Troy, University of Pittsburgh
    "Cooling a solid to ground state"
     Abstract:
    A major goal of quantum computing research is to drain all quanta (q) of thermal energy from a solid at a positive temperature T0 > 0, leaving the object in its ground state. In 2010 the first complete success was reported when a quantum drum was cooled to its ground state at T0 = 20mK. However, current theory, which is based on the Bose-Einstein equation, predicts that temperature T goes to 0 as q goes to 0. We prove that this discrepancy between experiment and theory is due to previously unobserved errors in low temperature predictions of the Bose-Einstein equation. We correct this error and derive a new formula for temperature which proves that T goes to T0 as q goes to 0. Simultaneously, the energy decreases to its 'supersolid' ground state level as q goes to 0+. For experimental data our temperature formula predicts that T0 = 9.8mK, in close agreement with the 20mK experimental result. Our results form a first step towards bridging the gap between existing theory and the construction of useful quantum computing devices.

  • Wednesday, October 26, 2011
    9:00 AM, Thackeray 703
     Michael Neilan, University of Pittsburgh
    "Finite element methods for singular perturbed PDEs"
     Abstract:
    In this talk, we discuss various theoretical and practical issues of solving two different singular perturbation problems. The first problem is a fourth order biharmonic problem that degenerates to Poisson's equation as the perturbation parameter tends to zero. Therefore convergent finite element methods must be convergent for both the biharmonic problem as well as Poisson's equation. We construct such methods by enriching (locally) Lagrange elements with volume and edge bubbles. The second problem that we discuss is the Brinkman problem, a linear Stokes equation when the perturbation parameter is large, but degenerates to a mixed formulation of Poisson's equation when the parameter is small. As such, numerical methods that work well for Stokes behave poorly when the perturbation parameter is small, where as methods that are designed for the mixed formulation of Poisson's equation do not converge when the parameter is large. We discuss how to augment H(div) elements with divergence free bubble functions to construct a family of stable and convergent methods for the Brinkman problem. Furthermore, using exact sequences of function spaces (discrete de Rham complexes) we show how these two seemingly unrelated problems are closely connected. Finally, if time permits we show how to extend this methodology to obtain conforming and divergence free elements for the Stokes/Brinkman problem on general triangular meshes.
    This is joint work with Johnny Guzm\'an (Brown) and Dmitriy Leykekhman (UConn).




    Last updated: October 19, 2011