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The melting and subsequent extraction of molten materials from the Earth's mantle is the subject of much scrutiny, yet remains poorly understood. Volcanism and magmatism observed at the Earth's surface are due to the partial melting of the convecting mantle and the segregation of melt from its parent solid. The formation of iron-rich cores in planetesimals are the result of segregation of dense metal melts, extracted from the primitive silica rich building blocks early in planetary evolution. The process of partial melting and melt segregation leads to the continual generation of the oceanic and continental crust, formation of metal-rich cores, and therefore dictates the large-scale chemical evolution of the silicate Earth and terrestrial planets.

Numerical solutions to fluid flow problems involve solving the linear systems arising from the discretization of the Stokes equation or a variant of it, which often have a saddle point structure and are difficult to solve. Geometric multigrid is a parallelizable method that can efficiently solve these linear systems especially for a large number of unknowns. We consider two approaches to solve these linear systems using geometric multigrid: First, we use a block preconditioner and apply geometric multigrid as in inner solver to the velocity block only. We develop deal.II tutorial step-56 to compare the use of geometric multigrid to other popular alternatives. This method is found to be competitive in serial computations in terms of performance and memory usage. Second, we design a special smoother to apply multigrid to the whole linear system. This smoother is analyzed as a Schwarz method using conforming and inf-sup stable discretization spaces. The resulting method is found to be competitive to a similar multigrid method using non-conforming finite elements that were studied by Kanschat and Mao. This approach has the potential to be superior to the first approach. Finally, expanding on the research done by Dannberg and Heister, we explore the analysis of a three-field Stokes formulation that is used to describe melt migration in the earth's mantle. Multiple discretizations were studied to find the best one to use in the ASPECT software package. We also explore improvements to ASPECT's linear solvers for this formulation utilizing block preconditioners and algebraic multigrid.

The melting and subsequent extraction of molten materials from the Earth's mantle is the subject of much scrutiny, yet remains poorly understood. Volcanism and magmatism observed at the Earth's surface are due to the partial melting of the convecting mantle and the segregation of melt from its parent solid. The formation of iron-rich cores in planetesimals are the result of segregation of dense metal melts, extracted from the primitive silica rich building blocks early in planetary evolution. The process of partial melting and melt segregation leads to the continual generation of the oceanic and continental crust, formation of metal-rich cores, and therefore dictates the large-scale chemical evolution of the silicate Earth and terrestrial planets.

Melting and melt segregation are key processes in the geochemical evolution of the Earth. However, mechanism and time scale of melt transport from the source to the surface are still not well understood and are dependent on the grain-scale distribution of melt. A related question is the retention of melt in partially molten regions of the Earths upper mantle. Seismic observations from mid-ocean ridges (MOR) and subduction zones are interpreted to show in-situ melt contents up to 3%, while geochemical observations from MOR basalts are inferred to indicate very efficient extraction of melt (porosities of order 0.1%). Earlier theoretical models of the melt distribution were based on the balance of surface tension between melt and uniform crystalline grains, predicting a simple net- work of melt along three-grain edges. Analyses of experimentally produced samples of olivine and basaltic melt show that the melt geometry is much more complex, and includes wetted two-grain boundaries. I reconstructed the melt geometry of two experimentally produced samples by serial sectioning and 3-D rendering of the pore geometry which demonstrates for the first time that melt exists in thin layers on two-grain boundaries. This confirms the inferences from previous 2-D observations and has significant implications for physical properties of partially molten regions, for example seismic velocities and attenuation. The wetted two-grain boundaries are inferred to be a consequence of continuous grain growth. Due to the complexity of the 3-D melt geometry the perme- ability of partially molten rocks can not be predicted from simple models. I therefore investigated the permeability as a function of porosity for both synthetic and ex- perimentally determined pore geometries using a lattice-Boltzmann method. The calculated permeability is not a simple function of porosity, but increases rapidly at a critical fraction of wetted two-grain boundaries. In order to extrapolate the experimentally based findings to grain sizes expected in natural rocks I examined the geometry of secondary phases inferred to represent relict melt in mantle peridotites from the Krivaja massif in Bosnia. These findings corroborate the experimental observations of wetted two-grain boundaries.