Iterative methods for the solution of linear systems are a core component of many scientific software packages, especially of numerical simulations in which the discretization of partial differential equations in two or three dimensions and with high spatial resolution often results in large, sparse linear systems. Since their introduction decades ago, iterative approaches such as the conjugate gradient method, GMRES, and BiCGStab have remained popular and relevant, and they have proven themselves to be scalable tools in eras of exponential growth in computing power and increasing heterogeneity of computing hardware.

In this thesis, I evaluate the convergence and computational performance of iterative solvers for linear systems obtained from modal discontinuous Galerkin (DG) discretizations. Specifically, I focus on systems that arise in the simulation of pore-scale multi-phase flow. Besides standard Krylov subspace methods with algebraic preconditioners, the evaluation focuses on multigrid methods, which, in their more algebraic variants, can also be viewed as iterative linear solvers. In particular, I discuss how p-multigrid methods, which use discretizations of different order instead of discretizations on different meshes, assume a simple algebraic structure for modal DG discretizations. I then show how hierarchical scale separation (HSS), a recently proposed multiscale method for modal DG discretizations, can be incorporated into the p-multigrid framework, and I discuss the unified implementation of both methods in highly parallel computing environments. I analyze how the computational performance of these methods is affected by their tunable parameters, and I demonstrate in numerical experiments that properly calibrated p-multigrid methods can accelerate large pore-scale flow simulations significantly. Moreover, I show how the main ideas of HSS techniques can be used to further accelerate p-multigrid methods.