Adaptive hp-Finite Element Methods for Stokes and Navier Stokes Equations
Adaptive hp-Finite Element Methods for Stokes and Navier Stokes Equations
The talk will discuss the components needed to produce an adaptive hp finite element method for Stokes and Navier Stokes flow, including the Babuska-Brezzi conditions for various velocity-pressure combinations, a posteriori error estimation and efficient domain decomposition methods for solving the resulting systems of equations. Numerical examples will be presented illustrating the theoretical results.
Computation of fourth order nonlinear parabolic PDEs
In recent years several mathematical models in fluid dynamics, materials science and plasticity have lead to fourth order nonlinear degenerate parabolic equations. We mention for example the lubrication approximation for thin viscous films
$$ u_{t} + \del .(\,b(u)\, \del \lap u) = 0, $$
where generically $b(u) := |u|^p$ for any given $p \in (0, \infty)$ and the Cahn-Hilliard equation
$$ \textstyle \frac{\partial u}{\partial t}= \del .(\,b(u)\, \del (-\gamma\lap u+\Psi'(u))) , $$
where $0\leq b_{\rm min}\leq b(\cdot)\leq b_{\rm max}$ is a diffusional mobility and $\Psi(\cdot)$ is a homogeneous free energy.
In this talk we analyse some fully practical piecewise linear finite element approximations and the resulting iterative schemes for solving the nonlinear discrete system. Also some numerical experiments are presented.
Papers describing this and related work can be accessed via the WWW addresses
http://link.springer.de/link/service/journals/00211/bibs/8080004/80800525.htm
http://fourier.dur.ac.uk:8000/~dma0jfb/Post/97-03.ps.Z
http://fourier.dur.ac.uk:8000/~dma0jfb/Post/97-07.ps.Z
http://fourier.dur.ac.uk:8000/~dma0jfb/Post/99-02.ps.Z
Integral equation methods for scattering by rough surfaces
We consider some of the numerical analysis and computational problems involved in predicting time harmonic wave scattering by rough surfaces. By a rough surface we mean a surface which: is approximately planar; is infinite in extent (or, at least, with diameter $>>\lambda$, where $\lambda$ is the wavelength); has local perturbations across the whole extent of the boundary which are $O(\lambda)$ in diameter. Examples are outdoor propagation of acoustic, radar, and radio waves over ground and sea surfaces, microwave scattering from, e.g., building and road surfaces, etc.
We discuss the formulation of these problems as boundary integral equations and the stability and convergence of the truncation of the domain that is required; numerical analysis of discretisation schemes with bounds which are uniform with respect to the size of the truncation; a matrix compression technique based on Chebyshev interpolation of the kernel in the direction orthogonal to the plane; iterative solution techniques for the resulting linear system. These aspects are illustrated by various numerical results.
Computational questions in mathematical biology
Mathematical biology depends heavily on scientific computing, but is not a traditional source of problems for numerical analysts. Therefore there are computational questions raised in mathematical biology that represent significant numerical challenges, and I will discuss a selection of these. I will begin by discussing some recent partial differential equation models. This will include models for ecological invasion in cyclical populations, where different numerical schemes have been shown to have very different convergence properties, and models for directed cell movement, which have a mixed hyperbolic--parabolic character. I will then move on to considering some new types of models in cell biology that reflect the intrinsic discreteness of a biological cell. I will focus on a model for scar tissue formation, and will show that this new type of model raises a range of important computational questions.