Past Computational Mathematics and Applications Seminar

9 November 2000
14:00
Dr Ian Sobey
Abstract
Boundary layers are often studied with no pressure gradient or with an imposed pressure gradient. Either of these assumptions can lead to difficulty in obtaining solutions. A major advance in fluid dynamics last century (1969) was the development of a triple deck formulation for boundary layers where the pressure is not specified but emerges through an interaction between boundary layer and the inviscid outer flow. This has given rise to new computational problems and computations have in turn fed ideas back into theoretical developments. In this survey talk based on my new book, I will look at three problems: flow past a plate, flow separation and flow in channels and discuss the interaction between theory and computation in advancing boundary layer theory.
  • Computational Mathematics and Applications Seminar
2 November 2000
14:00
Dr David Silvester
Abstract
This talk reviews some theoretical and practical aspects of incompressible flow modelling using finite element approximations of the (Navier-) Stokes equations. The infamous Q1-P0 velocity/pressure mixed finite element approximation method is discussed. Two practical ramifications of the inherent instability are focused on, namely; the convergence of the approximation with and without regularisation, and the behaviour of fast iterative solvers (of multigrid type) applied to the pressure Poisson system that arises when solving time-dependent Navier-Stokes equations using classical projection methods. \\ \\ This is joint work with David Griffiths from the University of Dundee.
  • Computational Mathematics and Applications Seminar
12 October 2000
14:00
Prof Howard Elman
Abstract
We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers.
  • Computational Mathematics and Applications Seminar
Dr Steven Benbow
Abstract
The talk will focus on solution methods for augmented linear systems of the form \\ \\ $[ A B ][x] = [b] [ B' 0 ][y] [0]$. \\ \\ Augmented linear systems of this type arise in several areas of numerical applied mathematics including mixed finite element / finite difference discretisations of flow equations (Darcy flow and Stokes flow), electrical network simulation and optimisation. The general properties of such systems are that they are large, sparse and symmetric, and efficient solution techniques should make use of the block structure inherent in the system as well as of these properties. \\ \\ Iterative linear solution methods will be described that attempt to take advantage of the structure of the system, and observations on augmented systems, in particular the distribution of their eigenvalues, will be presented which lead to further iterative methods and also to preconditioners for existing solution methods. For the particular case of Darcy flow, comments on properties of domain decomposition methods of additive Schwarz type and similarities to incomplete factorisation preconditioners will be made.
  • Computational Mathematics and Applications Seminar

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