Scottish Computational Mathematics Symposium

16th Annual Meeting
Tuesday 11th September 2007

Court-Senate Suite, Collins Building, Richmond Street, Strathclyde University, Glasgow

Speakers, titles and abstracts:

• Mike Baines (University of Reading)
  Fluid Meshes: Lagrangian Moving Mesh Finite Elements for Partial Differential Equations.
  Abstract: Moving mesh methods for fluid dynamics based on a Lagrangian formulation of the fluid equations have been in use since the 1960s, but their application has been restricted owing to the necessity of frequent re-meshing. The methods are often used in combination with a re-map step which returns the method to an Eulerian scheme, or alternatively the computed velocities are modified within an ALE (Arbitrary Lagrangian Eulerian) framework.

  Nevertheless, there are many modelling situations where a Lagrangian framework is useful, for example in nonlinear problems having moving boundaries or evolving singularities. Moreover, there are good mathematical reasons why moving meshes should be employed, for example to maintain the scale invariance of the original problem.

  It is only relatively recently that researchers have made a concerted effort to exploit moving mesh methods. An early method was the Moving Finite Element method of Miller, and this has been followed by MMPDEs (Moving Mesh Partial Differential Equations) based on equidistribution and its generalisations.

  In this talk I shall discuss the application of a Lagrangian moving mesh finite element method to moving boundary problems. The basic idea is to partition a conserved global quantity and maintain it in time (as in Lagrangian fluid dynamics). The method inherits the scale invariance of the original problem and can be generalised to include artificial densities (acting as monitor functions) and vorticities. The mesh velocities are generated via a distributed form of the Reynolds Transport Theorem.

  The advantages and limitations of the approach will be described.
  

• Gabriel Barrenechea (Strathclyde University: moving from Universidad de Cencepcion, Chille)
  A low-order mass conservative finite element method for the Darcy equation
  Abstract: In this talk we present a new finite element method for the Darcy equation. This method has the advantage of preserving nodal degrees of freedom for the velocity field, and providing a discrete velocity which is easily post-processed to become locally mass-conservative. The method is constructed starting from an enriched version of the lowest order P1/P0 pair and building enrichment functions for the velocity and pressure, which are statically condensed, thus not adding extra degrees of freedom to the computation. Stability and convergence results are proved for the method and several numerical results are given in order to confirm the theoretical results.
  
  
• Catherine Powell (University of Manchester)
  Efficient Solvers for Stochastic Finite Element Systems
  Abstract: In the last few years, interest in so-called stochastic finite element methods (SFEMs) has risen sharply within the numerical analysis and linear algebra communities. SFEMs are many and varied but all facilitate the solution of PDEs where uncertainties are present in material coefficients, boundary conditions and/or source terms. In this talk, we present an introduction to the so-called stochastic Galerkin formulation of elliptic PDEs with random field coefficients and focus on the efficient solution of the resulting linear systems. To illustrate the main ideas, the stochastic Darcy flow problem serves as a convenient example. Fluid flow in porous media is often modelled under the assumption that the conductivity coefficients are known at every spatial location. However, simulations based on such over-simplifications cannot provide quantification of probabilities of unfavourable events. SFEMs provide a framework for incorporating statistical information about spatial variability in the conductivity coefficients into computer simulations in such a way that comprehensive probabilistic information about the fluid velocity and pressure is obtained. Specifically, the conductivity coefficients and solution variables are represented as random fields. In contrast to the Monte Carlo method, where large numbers of deterministic problems are solved and then post-processed in order to learn about the statistical properties of the output variables, we discretise probability space directly and solve one large problem once. The key idea is to expand the random inputs, which might, say, be modelled as Gaussian or lognormal random fields with stated means and correlation functions, using suitable stochastic basis functions that respect the assumed statistics. After truncating the stochastic expansion, and discretising appropriately in space, we are required to solve a single system for the coefficients in the stochastic expansion of the solution. The moments of the numerical solutions are available from this data. Some SFEMs give rise to very large, coupled, systems of linear equations and consequently have suffered from bad press. We report on fast, robust linear algebra techniques and preconditioning schemes based on multigrid methods, for solving the stochastic Galerkin formulation of the Darcy flow problem. In primal form, this is relatively straight-forward. Using SFEMs in the context of solving stochastic mixed variational problems, however, is still a relatively new and unexplored field. The mixed formulation of the stochastic diffusion equation, which arises in groundwater flow modelling, and leads to the computation of a flow field, requires an approximation to the inverse of a random field coefficient. Preliminary results for solving the mixed problem via SFEMs, efficient solvers for the associated saddle-point systems, and a comparison with Monte Carlo methods, are presented.
  
  
• Jason Reese (University of Strathclyde)
  Beyond Navier-Stokes: computing the complex fluid mechanics of hypersonic aerodynamics and micro-scale gas flow
  Abstract: I will describe our ongoing work into the new fluid dynamics needed for very high-speed or micro- and nano-scale gas flows. While these are application areas of emerging technological importance, remarkably there is currently no sufficiently accurate and computationally efficient fluid dynamic model. This is because the conventional Navier-Stokes fluid equations do not allow for the local thermodynamic non-equilibrium that makes the behaviour of these rarefied gas flows uniquely complex. In this talk I will focus on our work on capturing two fundamental aspects of these flows: gas velocity slip at solid surfaces, and the associated Knudsen layer extending from the solid surface into the flow. I will describe the successes and failures of various hydrodynamic models (derived from kinetic theory) being developed to capture this non-equilibrium flow physics, and give examples of current test applications in both hypersonics and micro gas flow situations.
  
  
• Holger Wendland (University of Sussex)
  Meshfree Methods: Theory and Applications
  Abstract: Meshfree methods become more and more important for the numerical simulation of complex real-world processes. Compared to classical, mesh-based methods they have the advantage of being more flexible, in particular for higher dimensional problems and for problems, where the underlying geometry is changing. However, often, they are also combined with classical methods to form hybrid methods.

  In this talk, I will discuss meshfree, kernel based methods. After a short introduction along the lines of optimal recovery, I will concentrate on results concerning convergence orders and stability. After that I will address efficient numerical algorithms. Finally, I will present some examples, including one from fluid-structure interaction, which will demonstrate why these methods are currently becoming Airbus's preferred solution in Aeroelasticity.