Program

Flows in porous media have important applications such as geothermal energy production, enhanced oil recovery or sequestration of CO₂. In this series of three lectures we review the most important models ranging from groundwater flow and transport to compositional multiphase flows, then discuss accurate discretization methods, in particular discontinuous Galerkin schemes, and solution strategies such as fully-coupled and operator splitting methods as well as solvers for the subproblems including algebraic multigrid. The last lecture will put emphasis on high-performance computing aspects, in particular the implementation of higher-order discontinuous Galerkin methods.

Krylov subspaces form the basis of many modern numerical methods for computing matrix decompostions, for solving linear algebraic systems and eigenvalue problems, or for computing matrix functions. They occur in applications throughout science and engineering. This is not by accident, but due to the fundamental nature of the mathematical ideas that are encoded in these spaces. In fact, Krylov subspaces are closely linked with classical concepts associated with matrices and linear operators, including invariant subspaces, the minimal polynomial, or the Jordan decomposition. For the practical application of Krylov subspaces it is essential to generate well conditioned bases at a low cost. The analysis of this important challenge again involves classical mathematical topics, for example the structure of orthogonalization processes or the normality of operators.

This course will explore the mathematical theory of Krylov subspaces. While discussing the classical mathematics associated with these spaces, we will discover some surprising links to other areas of science.

Probability measures on Hilbert spaces provide a natural setting for random fields, understood as smooth random functions. Functions with varying degrees of smoothness arise as solutions of partial differential equations. Topics include the mean of Hilbert space-valued random elements as a weak integral, random elements with finite second moments, covariance, trace class operators, smoothness properties of random functions, Gaussian measures, Karhunen-Loeve expansion, and white noise distribution. The infinitely dimensional setting is useful as a limit case of high-dimensional problems, arising, e.g., when meshes in numerical models are refined, and it is at the foundation of the field of uncertainty quantification. Applications include Bayesian approach to inverse problems with infinitely dimensional data and assimilation of active fires detection from satellites into a fire spread model.

Over geological time, the solid mantle of the Earth behaves as a creeping fluid, undergoing a thermochemical convection that has plate tectonics as a surface expression. The mantle is modelled as an incompressible fluid with a very high viscosity. Volcanism is another surface expression of mantle convection: the liquid magma is derived from partial melting of the convecting mantle. Magma that forms by partial melting migrates through the interconnected pore-spaces between mantle grains. Hence the magma/mantle system is a two-phase flow: both the low-viscosity magma and the high-viscosity mantle can be modelled as low-Reynolds number flows. Since the late 1970s, a mathematical theory of two-phase flow been developed and extensively studied, both analytically and numerically. In this series of three lectures, I review the theory and provide some examples of its applications.

Lecture 1: foundations. Discussion of the physics of mantle convection, melting, and melt transport. Derivation of the mechanical governing equations from conservation of mass and momentum. Non-dimensionalisation and analysis of emergent length-scale.
Lecture 2: fundamentals. Simple solutions for two-phase flow at plate tectonic boundaries. Fluid-dynamical instabilities in the magma/mantle system: mechanical porosity-banding instabilities; reactive channelisation instabilities.
Lecture 3: applications. Thermo-chemistry of mantle melting. Melt production at mid-ocean ridges. Consequences of mantle heterogeneity. Consequences of grain-size variation.