ABSTRACT: Iterative linear solvers are a core component of numerical simulations in which the discretization of partial differential equations results in large, sparse linear systems. Since their introduction decades ago, iterative methods, such as Krylov subspace methods and multigrid techniques, have remained popular and relevant, and they have proven to be scalable tools in eras of exponential growth in computing power and increasing heterogeneity of computing hardware.
In this presentation, we explore the specific structure of linear systems in discontinuous Galerkin (DG) discretizations with modal bases, and how this structure relates to a hierarchy of DG discretizations of different order. p-Multigrid and hierarchical scale separation (HSS) methods exploit this hierarchy in order to accelerate the solution of these linear systems. I show how HSS can be interpreted as a p-multigrid method, and I discuss the parallel implementation of both approaches in a unified framework. The convergence and computational performance of both methods are evaluated in numerical experiments with large-scale simulations of two-phase flow in porous media