Dmitri Kuzmin - University of Dortmund

Discrete maximum principles and a posteriori error estimation for finite element approximations to transport equations

January 14, 2009
3pm, 104 SEC

Abstract

Continuous and discrete maximum principles are formulated for scalar transport equations. Relevant a priori bounds are enforced within the framework of algebraic flux correction. A family of nonlinear high-resolution finite element schemes is presented and combined with adaptive mesh refinement. The derivation of computable error indicators is discussed in some detail. Gradient recovery techniques are revisited and integrated into a new goal-oriented a posteriori error estimate. The methodology to be presented builds on the duality argument and features a node-based approach to localization of errors in the quantity of interest, as represented by a linear target functional. A possible violation of Galerkin orthogonality is taken into account in a simple and natural way. The use of an averaged gradient makes it possible to obtain a nonoscillatory distribution of weighted residuals without introducing jump terms. The weights are determined using the difference between the linear and quadratic finite element interpolants of the dual solution. The benefits of mesh adaptation are illustrated by numerical results for scalar conservation laws and hyperbolic systems.

Mikhail Perepelitsa - Vanderbilt University

Cusp Formation in 2-D Fluid Flows

Thursday, January 29, 2009
3-4 PM, 634 SR1

Abstract

We’ll consider the dynamics of localized vortex patches in the model of the Euler equations for 2-D incompressible, inviscid fluid flows. The solutions of the Euler equations describing such motion, as well as the flow maps they generate, have limited regularity but nevertheless are uniquely defined for all times [V.I. Youdovich ’63]. Moreover, the boundaries of initially smooth vortex patches retain their structure [J.-Y. Chemin ’93]. Results of numerical simulation have been reported that show that a corner singularity in the boundary of a vortex patch evolves into a cusp [A. Cohen, R. Danchin ’00]. We'll give an analytical proof of this observation and show, in fact, that the cusp forms instantaneously. This phenomenon can also be traced in the motion of compressible flows, modeled by solutions of the Navier-Stokes equations. This is joint work with David Hoff (Indiana University).

Eric Vanden-Eijnden - The Courant Institute

Transition Pathways of Rare Reactive Events in Complex Systems.

February 04, 2009
location: 204 SEC
time: 3:00pm -4:00 pm

Abstract

The dynamics of biomolecular systems is typically characterized by a wide range of time scales, complicating their study via computer simulations. Of particular difficulty are situations which involve rare reactive events such as conformation changes of macromolecules, nucleation events during first-order phase transitions, chemical reactions, or bistable behavior of genetic switc. The occurrence of these rare events is related to the presence of dynamical bottlenecks of energetic and/or entropic origin which effectively partition the configuration space of the system into metastable basins. The system spends most of its time fluctuating within these long-lived metastable states and only rarely makes transitions between them. The rare events then determine the long-time evolution of the system. In this talk, I will present a general theoretical framework termed transition path theory (TPT) for the description of rare reactive events and compare it to other approaches such as the classical transition state theory (TST) and the more recent transition path sampling (TPS). I will also show that TPT can used to design efficient numerical algorithms such as the string method for the identification of the pathway, free energy and rate of the rare events. Both the theory and the numerics will be illustrated via examples.

Margaret Beck - Faculty Candidate

Nonlinear stability of time-periodic viscous shocks.

February 9, 2009
301 AH
3:00pm -4:00 pm

Abstract

If a given solution of a PDE is stable, then, roughly speaking, any other solution that starts near it, stays near it for all time. This is an important concept in applications, because it is typically only the stable solutions that are observed in practice. I will outline two key mathematical difficulties that one can encounter when analyzing the stability of time-periodic solutions of dissipative PDEs on unbounded domains. Briefly, they are the presence of zero eigenvalues that are embedded in the continuous spectrum and the time-periodicity of the associated linear operator. In the context of viscous shocks in systems of conservation laws, I will show how these difficulties can be overcome. The method involves the development of a contour integral representation of the linear evolution, similar to that of a strongly continuous semigroup, and detailed pointwise estimates on the resultant Greens function, which are sufficient for proving nonlinear stability under the necessary assumption of spectral stability.

Yuliya Gorb - Faculty Candidate

Multiscale Modeling and Simulation of Fluid Flows in Deformable Porous Media

February 12, 2009
PGH646
11:00am -12:00 am

Abstract

The main focus of this talk is on fluid flows in deformable elastic media and associated multiscale problems. Many upscaling methods are developed for flows in rigid porous media or deformable elastic media assuming infinitely small fluid-solid interface displacements relative to the pore size. Much research is needed for the most general and least studied problem of flow in deformable porous media when the fluid-solid interface deforms considerably at the pore level. We introduce a general framework for numerical upscaling of the deformable porous media in the context of a multiscale finite element method. This method allows for large interface displacements and significant changes in pore geometry and volume. For linear elastic solids we present some analysis of the proposed method.

L. Mahadevan - Harvard University

On growth and form: mathematics, mechanics and morphogenesis

February 25, 2009
3pm, 204 SEC

Abstract

The growth and form of a soft solid pose a range of problems that combine aspects of geometry and physics. I will discuss some examples of growth and form in the plant and animal world motivated by qualitative and quantitative biological observations. The problems include the shape of a freely growing pollen tube, the undulating fringes on a leaf or petal, the folds in a mammalian brain, and the coiling of a vertebrate intestine. In each case, we will see how a combination of physical experiments, mathematical models and simple computations allow us to unravel the basis for the diversity and complexity of shape in biology.

Analisa Quaini- Faculty Candidate

Algorithms for fluid-structure interaction problems arising in hemodynamics

March 04, 2009
646 PGH
3:00am -4:00 am

Abstract

The main focus of this talk is on fluid flows in deformable elastic media and associated multiscale problems. Many upscaling methods are developed for flows in rigid porous media or deformable elastic media assuming infinitely small fluid-solid interface displacements relative to the pore size. Much research is needed for the most general and least studied problem of flow in deformable porous media when the fluid-solid interface deforms considerably at the pore level. We introduce a general framework for numerical upscaling of the deformable porous media in the context of a multiscale finite element method. This method allows for large interface displacements and significant changes in pore geometry and volume. For linear elastic solids we present some analysis of the proposed method.

Yi Sun - Faculty Candidate

Network dynamics of Hodgkin-Huxley neurons

March 05, 2009
646 PGH
04:00pm -05:00pn

Abstract

The reliability and predictability of neuronal network dynamics is a =20 central question in neuroscience. We present a numerical analysis of =20 the dynamics of all-to-all pulsed-coupled Hodgkin-Huxley (HH) neuronal =20 networks. Since this is a non-smooth dynamical system, we propose a =20 pseudo-Lyapunov exponent (PLE) that captures the long-time =20 predictability of HH neuronal networks. The PLE can capture very well =20 the dynamical regimes of the network. Furthermore, we present an =20 efficient library-based numerical method for simulating HH neuronal =20 networks. Our pre-computed high resolution data library can allow us =20 to avoid resolving the spikes in detail and to use large numerical =20 time steps for evolving the HH neuron equations. By using the =20 library-based method, we can evolve the HH networks using time steps =20 one order of magnitude larger than the typical time steps used for =20 resolving the trajectories without the library, while achieving =20 comparable resolution in statistical quantifications of the network =20 activity. Moreover, our large time steps using the library method can =20 overcome the stability requirement of standard ODE methods for the =20 original dynamics. Our method may be applied to other biological =20 networks which evolve stiff dynamics.