Past Computational Mathematics and Applications Seminar

7 March 2002
14:00
Dr Alison Ramage and Prof Howard Elman
Abstract
It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. For the Galerkin finite element method with linear elements on a uniform 1D grid, a precise statement as to exactly when such oscillations occur can be made, namely, that for a problem with mesh size h, constant advective velocity and different values at the left and right boundaries, oscillations will occur if the mesh P\'{e}clet number $P_e$ is greater than one. In 2D, however, the situation is not so well understood. In this talk, we present an analysis of a 2D model problem on a square domain with grid-aligned flow which enables us to clarify precisely when oscillations occur, and what can be done to prevent them. We prove the somewhat surprising result that there are oscillations in the 2D problem even when $P_e$ is less than 1. Also, we show that there are distinct effects arising from differences in the top and bottom boundary conditions (equivalent to those seen in 1D), and the non-zero boundaries parallel to the flow direction.
  • Computational Mathematics and Applications Seminar
Abstract
Algebra based modeling systems are becoming essential elements in the application of large and complex mathematical programs. These systems enable the abstraction, expression and translation of practical problems into reliable and effective operational systems. They provide the bridged between algorithms and real world problems by automating the problem analysis and translation into specific data structures and provide computational services required by different solvers. The modeling system GAMS will be used to illustrate the design goals and main features of such systems. Applications in use and under development will be used to provide the context for discussing the changes in user focus and future requirements. This presents new sets of opportunities and challenges to the supplier and implementer of mathematical programming solvers and modeling systems.
  • Computational Mathematics and Applications Seminar
14 February 2002
14:00
Dr Roland Becker
Abstract
A systematic approach to error control and mesh adaptation for optimal control of systems governed by PDEs is presented. Starting from a coarse mesh, the finite element spaces are successively enriched in order to construct suitable discrete models. This process is guided by an a posteriori error estimator which employs sensitivity factors from the adjoint equation. We consider different examples with the stationary Navier-Stokes equations as state equation.
  • Computational Mathematics and Applications Seminar
Prof Ivan Graham
Abstract
When steady solutions of complex physical problems are computed numerically it is often crucial to compute their <em>stability</em> in order to, for example, check that the computed solution is "physical", or carry out a sensitivity analysis, or help understand complex nonlinear phenomena near a bifurcation point. Usually a stability analysis requires the solution of an <em>eigenvalue problem</em> which may arise in its own right or as an appropriate linearisation. In the case of discretized PDEs the corresponding matrix eigenvalue problem will often be of generalised form: \\ $Ax=\lambda Mx$ (1) \\ with $A$ and $M$ large and sparse. In general $A$ is unsymmetric and $M$ is positive semi-definite. Only a small number of "dangerous" eigenvalues are usually required, often those (possibly complex) eigenvalues nearest the imaginary axis. In this context it is usually necessary to perform "shift-invert" iterations, which require repeated solution of systems of the form \\ $(A - \sigma M)y = Mx$, (2) \\ for some <em>shift</em> $\sigma$ (which may be near a spectral point) and for various right-hand sides $x$. In large applications systems (2) have to be solved iteratively, requiring <em>"inner iterations"</em>. \\ \\ In this talk we will describe recent progress in the construction, analysis and implementation of fast algorithms for finding such eigenvalues, utilising algebraic domain decomposition techniques for the inner iterations. \\ \\ In the first part we will describe an analysis of inverse iteration techniques for (1) for a model problem in the presence of errors arising from inexact solves of (2). The delicate interplay between the convergence of the (outer) inverse iteration and the choice of tolerance for the inner solves can be used to determine an efficient iterative method provided a good preconditioner for $A$ is available. \\ \\ In the second part we describe an application to the computation of bifurcations in Navier-Stokes problems discretised by mixed finite elements applied to the velocity-pressure formulation. We describe the construction of appropriate preconditioners for the corresponding (3 x 3 block) version of (2). These use additive Schwarz methods and can be applied to any unstructured mesh in 2D or 3D and for any selected elements. An important part of the preconditioner is the adaptive coarsening strategy. At the heart of this are recent extensions of the Bath domain decomposition code DOUG, carried out by Eero Vainikko. \\ \\ An application to the computation of a Hopf bifurcation of planar flow around a cylinder will be given. \\ \\ This is joint work with Jörg Berns-Müller, Andrew Cliffe, Alastair Spence and Eero Vainikko and is supported by EPSRC Grant GR/M59075.
  • Computational Mathematics and Applications Seminar
Dr Milan Mihajlovic
Abstract
In this presentation we examine the convergence characteristics of a Krylov subspace solver preconditioned by a new indefinite constraint-type preconditioner, when applied to discrete systems arising from low-order mixed finite element approximation of the classical biharmonic problem. The preconditioning operator leads to preconditioned systems having an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. We compare the convergence characteristics of a new approach with the convergence characteristics of a standard block-diagonal Schur complement preconditioner that has proved to be extremely effective in the context of mixed approximation methods. \\ \\ In the second part of the presentation we are concerned with the efficient parallel implementation of proposed algorithm on modern shared memory architectures. We consider use of the efficient parallel "black-box'' solvers for the Dirichlet Laplacian problems based on sparse Cholesky factorisation and multigrid, and for this purpose we use publicly available codes from the HSL library and MGNet collection. We compare the performance of our algorithm with sparse direct solvers from the HSL library and discuss some implementation related issues.
  • Computational Mathematics and Applications Seminar

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