Computational Mathematics and Applications

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.

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.

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.