Comlab

The use of discrete differential forms in the construction of finite element discretisations of the Sobolev spaces H^s, H(div) and H(curl) is now routinely applied by numerical analysts and engineers alike. However, little attention has been paid to the conditioning of the resulting stiffness matrices, particularly in the case of the non-uniform meshes that arise when adaptive refinement algorithms are used. We study this issue and show that the matrices are generally rather poorly conditioned. Typically, diagonal scaling is applied (often unwittingly) as a preconditioner. However, whereas diagonal scaling removes the effect of the mesh non-uniformity in the case of Sobolev spaces H^s, we show this is not so in the case of the spaces H(curl) and H(div). We trace the reason behind this difference, and give a simple remedy for curing the problem.

Non-normal matrices can be tiresome; some eigenvalues may be phlegmatic while others may be volatile. Computable error bounds are rarely used in such computations. We offer a way to proceed. Let (e,q,p') be an approximate eigentriple for non-normal B. Form column and row residuals r = Bq - qe and s' = p'B - ep'. We establish the relation between the smallest perturbation E, in both spectral and Frobenius norms, that makes the approximations correct and the norms of r and s'. Our results extend to the case when q and p are tall thin matrices and e is a small square matrix. Now regard B as a perturbation of B-E to obtain a (first order) bound on the error in e as a product of ||E|| and the condition number of e, namely (||q|| ||p'||)/|p'q|.

We present a new stabilized mixed finite element method for the linear elasticity problem. The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive and equilibrium equations, and from the relation defining the rotation in terms of the displacement.

We show that the resulting augmented variational formulation and the associated Galerkin scheme are well posed, and that the latter becomes locking-free and asymptotically locking-free for Dirichlet and mixed boundary conditions, respectively. In particular, the discrete scheme allows the utilization of Raviart-Thomas spaces of lowest order for the stress tensor, piecewise linear elements for the displacement, and piecewise constants for the rotation.

In the case of mixed boundary conditions, the essential one (Neumann) is imposed weakly, which yields the introduction of the trace of the displacement as a suitable Lagrange multiplier. This trace is then approximated by piecewise linear elements on an independent partition of the Neumann boundary whose mesh size needs to satisfy a compatibility condition with the mesh size associated to the triangulation of the domain. A reliable and efficient a-posteriori error estimate is also described. Finally, several numerical results illustrating the performance of the augmented scheme are reported.

The Cahn-Hilliard equations provides a model of phase transitions when two or more immiscible fluids interact. When coupled with the Navier-Stokes equations we obtain a model fro the dynamics of multiphase flow. This model takes into account the viscosity and densities of the various fluids present.

A finite element discretisation of the variable density Cahn-Hilliard-Navier-Stokes equations is presented. An analysis of the discretisation and a reliable efficient numerical solution method are presented.

I will describe some application areas for radial basis function, and discuss how the computational problems can be overcome by the use of preconditioning methods and fast evaluation techniques.

This Seminar has been cancelled and will now take place in Trinity Term, Week 3, 11 MAY.

Using the one-dimensional diffusion equation as an example, this seminar looks at ways of constructing approximations to the solution and coefficient functions of differential equations when the coefficients are not fully defined. There may, however, be some information about the solution. The input data, usually given as values of a small number of functionals of the coefficients and the solution, is insufficient for specifying a well-posed problem, and so various extra assumptions are needed. It is argued that looking at these inverse problems as problems in Bayesian statistics is a unifying approach. We show how the standard methods of Tikhonov Regularisation are related to special forms of random field. The numerical approximation of stochastic partial differential Langevin equations to sample generation will be discussed.

Standard finite element or boundary element methods for high frequency scattering problems, with piecewise polynomial approximation spaces, suffer from the limitation that the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency. Here we present a new boundary element method for which, by including in the approximation space the products of plane wave basis functions with piecewise polynomials supported on a graded mesh, we can demonstrate a computational cost that grows only logarithmically with respect to the frequency.