Past Computational Mathematics and Applications Seminar

Abstract

We combine linear algebra techniques with finite element techniques to obtain a reliable stopping criterion for the Conjugate Gradient algorithm. The finite element method approximates the weak form of an elliptic partial differential equation defined within a Hilbert space by a linear system of equations A x = b, where A is a real N by N symmetric and positive definite matrix. The conjugate gradient method is a very effective iterative algorithm for solving such systems. Nevertheless, our experiments provide very good evidence that the usual stopping criterion based on the Euclidean norm of the residual b - Ax can be totally unsatisfactory and frequently misleading. Owing to the close relationship between the conjugate gradient behaviour and the variational properties of finite element methods, we shall first summarize the principal properties of the latter. Then, we will use the recent results of [1,2,3,4]. In particular, using the conjugate gradient, we will compute the information which is necessary to evaluate the energy norm of the difference between the solution of the continuous problem, and the approximate solution obtained when we stop the iterations by our criterion.

Finally, we will present the numerical experiments we performed on a selected ill-conditioned problem.

References

  • [1] M. Arioli, E. Noulard, and A. Russo, Vector Stopping Criteria for Iterative Methods: Applications to PDE's, IAN Tech. Rep. N.967, 1995.
  • [2] G.H. Golub and G. Meurant, Matrices, moments and quadrature II; how to compute the norm of the error in iterative methods, BIT., 37 (1997), pp.687-705.
  • [3] G.H. Golub and Z. Strakos, Estimates in quadratic formulas, Numerical Algorithms, 8, (1994), pp.~241--268.
  • [4] G. Meurant, The computation of bounds for the norm of the error in the conjugate gradient algorithm, Numerical Algorithms, 16, (1997), pp.~77--87.
  • 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

Pages