Abstract/Summary

Here we investigate finite element techniques aimed at preserving the underlying geometric structures for various problems, and, in doing so, develop new geometric structure preserving methods. We initially focus on systems of Hamiltonian ODEs, examining the place of existing methods as geometric numerical integrators. We then develop a new geometrical finite element method for Hamiltonian ODEs with a view to generalise it to be the temporal discretisation of a space-time adaptive finite element method. We go on to investigate how well finite element methods can preserve the structure of Hamiltonian PDEs, which are a large class of physically relevant PDEs possessing a conserved physical invariant, the Hamiltonian functional, which often physically represents the energy of the problem. Examples of this kind of problem include, but are not limited to, oceanographical models of wave propagation such as KdV type equations and the nonlinear Schr¨odinger equations, and the semi-geostrophic equations for atmospheric modelling. We construct a general methodology for the design of finite element schemes for such problems and go on to develop multiple schemes in this framework for not only Hamiltonian PDEs but also systems of Hamiltonian PDEs. Within the study of finite element methods for Hamiltonian PDEs we prove both a priori and a posteriori error bounds, in addition to examining the role of spatial adaptivity for our schemes.