Day 2 Titles and Abstracts



ADI finite element methods for an evolution equation with a positive type kernel

Graeme Fairweather
Colarado School of Mines, USA
gxf -at- ams.org

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For more than half a century, alternating direction implicit (ADI) methods have proved to be effective techniques in the solution of multidimensional time-dependent problems, since they reduce such a problem to the solution of systems of independent one-dimensional problems. We describe recent progress in the formulation and analysis of ADI methods for partial integro-differential equations (PIDEs) with a positive type kernel. In these methods, orthogonal spline collocation (also known as spline collocation at Gauss points) or the finite element Galerkin (FEG) method is used for the spatial discretization, and the time-stepping is done with ADI methods based on the Crank-Nicolson method or the second-order backward differentiation formula. In the case of the FEG, PIDEs with both smooth and nonsmooth kernels are considered. The methods are proved to be of optimal accuracy in the $ L^{\infty}(L^2)$-norm which is confirmed by results of numerical experiments. This is joint work with Ryan Fernandes, Morrakot Khebchareon and Amiya Pani.

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Oldroyd model of viscoelastic fluids: Some theoretical and computational issues

Amiya Kumar Pani
Indian Institute of Technology Bombay, India
akp -at- math.iitb.ac.in

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Newton's model of incompressible viscous fluid is described by a wellknown system of Navier-Stokes equations. This has been a basic model for describing flow at moderate velocities of majority of viscous incompressible fluids encountered in practice. However, models of viscoelastic fluids have been proposed in the mid twentieth century which take into account the prehistory of the flow and are not subject to Newtonian frame. One such model was proposed by J.G. Oldroyd and hence, it is then named after him. The equation of motion in this case gives rise to the following integro-differential equation

$\displaystyle \frac{\partial u}{\partial t} + u\cdot\nabla u - \mu\Delta u - \... ...(t-\tau)\Delta u(x,\tau)\,d \tau + \nabla p = f(x,t), \;x\in \Omega, t>0, (*) $

and incompressibility condition

$\displaystyle \bigtriangledown.u = 0,\;\; x\in\Omega, t>0$

with initial condition

$\displaystyle u(x,0)=u_0, u=0 , \; x\in \partial\Omega, t\geq 0.$

Here, $ \Omega$ is a bounded domain in $ R^{2}$ with boundary $ \partial\Omega, \mu>0$ and the kernel $ \beta(t)=\gamma exp(-\delta t)$, where both $ \gamma$ and $ \delta$ are positive constants. With a brief discussion on the model, we, in the first part of this talk, concentrate on a recently derived uniform estimates in time whose proof was bothering us for the last 10 to 12 years. With a brief note on existence, we look into some new regularity results under realistically assumed conditions on the initial data. In the second part, we apply finite element Galerkin approximations to the above system and discuss convergence analysis without assuming compatibility conditions which are hard to verify while conducting numerical experiments. Since the problem (*) is an integral perturbation of the Navier-Stokes equations, we would like to discuss `how far results on finite element analysis for the Navier-Stokes equations can be carried over to the present case.' We conclude the talk with some theoretical and computational issues.

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Logarithmic decay of hyperbolic equations with arbitrary small boundary damping

Xiaoyu Fu
Indian Institute of Technology Bombay, India
rj_xy -at- 163.com

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This topic is addressed to an analysis on the longtime behavior of the hyperbolic equations with a partially boundary damping, under sharp regularity assumptions on the coefficients appeared in the equation. Based on a global Carleman estimate, we establish an estimate on the underlying resolvent operator of the equation, via which, we show the logarithmic decay rate for solutions of the hyperbolic equations without any geometric assumption on the subboundary in which the damping is effective.

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Regularity properties for the Hilbert Uniqueness Method for linear conservative equations and applications to convergence rates for discrete controls.

Sylvain Ervedoza
Innstitut de Mathématiques, Université Paul Sabitier,Toulouse University, France
ervedoza -at- math.univ-toulouse.fr

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(Joint work with Enrique Zuazua (Basque Center for Applied Mathematics)

Our goal is to present the Hilbert Uniqueness Method (HUM) introduced by Jacques-Louis Lions for computing controls for linear conservative systems (waves, Schrodinger, etc). The main idea is to solve a control problem using observability properties for the adjoint equation. This method is by now classical, but the smoothness properties of the control provided by this technique has not been studied precisely so far. For instance, one could expect that a smooth initial data to be controlled could be controlled with a smooth control. Though this is expected, it turns out that the control given by the classical HUM does not enjoy such nice properties, as we will see on a basic counterexample.

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