Networks of faults provide a tricky problem of representing discontinuous fields on complicated geometries. I've been working on using extended finite element methods that do not require the mesh to conform to the discontinuities by embedding the discontinuity in the basis function. The resulting methods allow for computation on even the nastiest of fault network geometries.

The dynamics of porous flow in highly heterogenous media are very dependent upon the small-scale heterogeneity. However, it is rarely computationally feasible to resolve the small scales while capturing the large simulation domains. Therefore, I've been working with colleagues to develop multilevel finite element upscaling methods to generate basis functions that respect sub-grid-scale heterogenaity.

Shear zones are perhaps the most obvious examples of localized deformation in geodynamics. I've been developing models of viscoelastic shear zones which localize as a result of grain size heterogeneity, demonstrating the importance of rheological considerations in localization. These models exhibit a wide range of dynamical phenomena, from periodic, catastrophic, earthquake-like events to multiple, distributed shear localizations.

The above work in shear zones has led to an interest in the interaction of the mantle and crust below slip-strike faults. The relative strengths of the upper mantle and lower crust is still in question, and this has interesting implications for the interaction of faults and viscous shear zones in the mantle below.