An important constraint on our ability to numerically simulate physical processes is our ability to solve the resulting linear systems. Multiscale methods, such as multigrid, provide optimal or near-optimal order solution techniques for a wide range of problems. Current multigrid methods (both geometric and algebraic) rely on the use of effective coarse-scale operators to achieve this efficiency. Classical geometric multigrid methods construct these operators based on the geometry of a given problem and are, thus, most effective when this geometry is known and simple. Algebraic multigrid methods are free from most geometric constraints, instead relying on assumptions about the character of the matrix that limit the applicability of such methods to a relatively small class of problems.

In this thesis, we first study the applicability of multigrid methods to the problem of flow through porous media, particularly motivated by the size of the linear systems resulting from discretization. For such problems, it is well known how to construct effective coarse-scale models for a correction-based multigrid method. For these robust multigrid solvers, we consider whether sufficient information is contained in the multigrid hierarchy to allow for efficient approximation of solutions of these linear systems. In particular, we examine important properties of the coarse-scale operators and the solutions to these coarse-scale equations and demonstrate their relevance to both the fine-scale and continuum-scale models.

Additionally, we consider the question of improvements to multigrid algorithms in the form of adaptivity or self-correction. By utilizing the multigrid method itself to expose slow-to-converge components, we may adapt the multigrid hierarchy in order to improve performance, resulting in improved efficiency for a new class of problems. We present results for both the classical algebraic multigrid and smoothed aggregation frameworks illustrating typical multigrid efficiency with fewer assumptions on the given matrix systems. We consider theoretical issues of this adaptive procedure in the reduction-based algebraic multigrid setting.