The implicit material point method for variable viscosity Stokes flow

Computational geodynamics is a rapidly expanding field which combines research from geology, geophysics, mathematics and computer science. With the availability of affordable, multi-core, highspeed computers, researchers are able to routinely run high resolution, geodynamic models in both two and three dimensions. The ever increasing computational resources enables researchers to explore new physical phenomena. In addition to the increase in computational geodynamic applications being explored, the recent surge of research in this area has spawned a constant stream of new numerical methods, suited for geodynamics, to be developed. Geodynamic processes present an interesting set of challenges for numerical models. The key requirements for developing geodynamic models is that they must be able to handle multiple materials, the evolution of the material must be able to be tracked whilst undergoing large deformations and the method must be able to deal with extremely large contrasts in material properties. In addition, the method should be able to use constitutive laws which are representative of rocks. Consequently, the method must permit rheologies which may be stress, strain rate or pressure dependent and possibly allow for a fully anisotropic constitutive relationship. Modelling approaches which have been demonstrated to address this set of requirements typically use, non-standard discretisations which employ both a mesh and a set of particles, or markers. Due to their hybrid nature, little numerical analysis has been performed to understand the theoretical behaviour of these methods. It is the purpose of this thesis to begin to bridge this gap and develop theory which supports the use of particle-mesh methods for studying geodynamics. We also draw on ideas from the finite element literature to develop a recovery technique for computing accurate values of stress, strain rate and pressure and to construct an efficient preconditioner to solve variable viscosity, Stokes flow problems