Multilevel Methods for Numerical Modeling with Applications in Hydrogeology

The project focuses on the development of new mathematical tools which improve our understanding and application of advanced computational methods. The research explores novel as well as established practical algorithms which are then validated and verified in a variety of hydrogeological scenarios. One major goal of computational mathematics research, in general, is to improve our understanding of numerical algorithms and to make them more efficient and accurate. However, the tasks of improving methods and then applying them are often completed by distinct groups of researchers and have long transfer times between development and application. Often, the theoretical findings and the heuristic algorithms developed by practitioners follow different trajectories. Indeed, many valuable numerical techniques have been proposed and used by practitioners without much theoretical justification or for a narrow set of problems. Concurrent improvements based on new mathematical insights are often unrelated to practical problems. This research addresses the disparity between the two disciplinary trajectories by reconnecting advanced and abstract mathematical theories with practice. The project has the potential to impact a wide range of applications, including for example the simulation of variably saturated flow.

This project is concerned with the development and analysis of adaptive, conservative and monotone discretizations that are extendable to any order for the solution of nonlinear partial differential equations, such as Richards' equation used to simulate variably saturated flow and Biot's model in poroelasticity. Typically, such discretization methods result in large-scale, ill-conditioned linear systems. The efficient solution of such systems, generally non-symmetric and indefinite, is crucial for the performance of overall numerical simulation as it consumes the larger part of the computing resources. Part of this project includes the development of a class of efficient and adaptive multilevel solvers capable of generating flexible hierarchies of spaces. Such hierarchies are useful also in filtering and representing sparse data sets, robust with respect to the structure and the type of data: smooth, oscillatory or combinations of the two. In the targeted applications in hydrogeology, the research will be on techniques which efficiently approximate elevation, groundwater head, precipitation, and other relevant hydrogeological spatial data.