Past

Discontinuous Galerkin methods (DG) use trial and test functions that are continuous within

elements and discontinuous at element boundaries. Although DG methods have been invented

in the early 1970s they have become very popular only recently.

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DG methods are very attractive for flow and transport problems in porous media since they

can be used to solve hyperbolic as well as elliptic/parabolic problems, (potentially) offer

high-order convergence combined with local mass balance and can be applied to unstructured,

non-matching grids.

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In this talk we present a discontinuous Galerkin method based on the non-symmetric interior

penalty formulation introduced by Wheeler and Rivi\`{e}re for an elliptic equation coupled to

a nonlinear parabolic/hyperbolic equation. The equations cover models for groundwater flow and

solute transport as well as two-phase flow in porous media.

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We show that the method is comparable in efficiency with the mixed finite element method for

elliptic problems with discontinuous coefficients. In the case of two-phase flow the method

can outperform standard finite volume schemes by a factor of ten for a five-spot problem and

also for problems with dominating capillary pressure.

The standard airframe industry tool for flutter analysis is based

on linear potential predictions of the aerodynamics. Despite the

limitations of the modelling this is even true in the transonic

range. There has been a heavy research effort in the past decade to

use CFD to generate the aerodynamics for flutter simulations, to

improve the reliability of predictions and thereby reduce the risk

and cost of flight testing. The first part of the talk will describe

efforts at Glasgow to couple CFD with structural codes to produce

a time domain simulation and an example calculation will be described for

the BAE SYSTEMS Hawk aircraft.

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A drawback with time domain simulations is that unsteady CFD is still

costly and parametric searches to determine stability through the

growth or decay of responses can quickly become impractical. This has

motivated another active research effort in developing ways of

encapsulating the CFD level aerodynamic predictions in models which

are more affordable for routine application. A number of these

approaches are being developed (eg POD, system identification...)

but none have as yet reached maturity. At Glasgow effort has been

put into developing a method based on the behaviour of the

eigenspectrum of the discrete operator Jacobian, using Hopf

Bifurcation conditions to formulate an augmented system of

steady state equations which can be used to calculate flutter speeds

directly. The talk will give the first three dimensional example

of such a calculation.

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For background reports on these topics see

http://www.aero.gla.ac.uk/Research/CFD/projects/aeroelastics/pubs/menu…

We discuss estimating the growth exponents for positive solutions to the

random parabolic Anderson's model with small parameter k. We show that

behaviour for the case where the spatial variable is continuous differs

markedly from that for the discrete case.

The probabilistic approach to homogenization can be adapted to fully

degenerate situations, where irreducibility is insured from a Doeblin type

condition. Using recent results on weak sense Poisson equations in a

similar framework, obtained jointly with A. Veretennikov, together with a

regularization procedure, we prove the homogenization result. A similar

approach can also handle degenerate random homogenization.

It is known for elliptic problems with smooth coefficients

that the solution is smooth in the interior of the domain;

low regularity is only possible near the boundary.

The $hp$-version of the FEM allows us to exploit this

property if we use meshes where the element size grows

porportionally to the element's distance to the boundary

and the approximation order is suitably linked to the

element size. In this way most degrees of freedom are

concentrated near the boundary.

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In this talk, we will discuss convergence and complexity

issues of the boundary concentrated FEM. We will show

that it is comparable to the classical boundary element

method (BEM) in that it leads to the same convergence rate

(error versus degrees of freedom). Additionally, it

generalizes the classical FEM since it does not require

explicit knowledge of the fundamental solution so that

it is also applicable to problems with (smooth) variable

coefficients.

Consider a spectrally one-sided Levy process X and reflect it at

its past infimum I. Call this process Y. We determine the law of the

first crossing time of Y of a positive level a in terms of its

'scale' functions. Next we study the exponential decay of the

transition probabilities of Y killed upon leaving [0,a]. Restricting

ourselves to the case where X has absolutely continuous transition

probabilities, we also find the quasi-stationary distribution of

this killed process. We construct then the process Y confined in

[0,a] and prove some properties of this process.

A reveiw of results about spectral analysis of generators of

some stochastic lattice models (a stochastic planar rotators model, a

stochastic Blume-Capel model etc.) will be presented. Then I'll discuss new

results by R.A. Minlos, Yu.G. Kondratiev and E.A. Zhizhina concerning spectral

analysis of the generator of stochastic continuous particle system. The

construction of one-particle subspaces of the generators and the spectral

analysis of the generator restricted on these subspaces will be the focus of

the talk.

Following the framework of Formaggia and Perotto (Numer.

Math. 2001 and 2003), anisotropic a posteriori error estimates have been

proposed for various elliptic and parabolic problems. The error in the

energy norm is bounded above by an error indicator involving the matrix

of the error gradient, the constant being independent of the mesh aspect

ratio. The matrix of the error gradient is approached using

Zienkiewicz-Zhu error estimator. Numerical experiments show that the

error indicator is sharp. An adaptive finite element algorithm which

aims at producing successive triangulations with high aspect ratio is

proposed. Numerical results will be presented on various problems such

as diffusion-convection, Stokes problem, dendritic growth.

Recently Friesecke, James and Muller established the following

quantitative version of the rigidity of SO(n) the group of special orthogonal

matrices. Let U be a bounded Lipschitz domain. Then there exists a constant

C(U) such that for any mapping v in the L2-Sobelev space the L^2-distance of

the gradient controlls the distance of v a a single roation.

This interesting inequality is fundamental in several problems concerning

dimension reduction in nonlinear elasticity.

In this talk, we will present a joint work with Muller and Zhong where we

investigate an analagous quantitative estimate where we replace SO(n) by an

arbitrary smooth, compact and SO(n) invariant subset of the conformal

matrices E. The main novelty is that exact solutions to the differential

inclusion Df(x) in E a.e.x in U are not necessarily affine mappings.