This thesis proposes the inexact hierarchical scale separation (IHSS) method for the solution of linear systems in modal discontinuous Galerkin (DG) discretizations. Like p-multigrid methods, IHSS alternates between discretizations of different polynomial order to improve the computational performance of solving linear systems. IHSS uses two discretizations, which are obtained from a hierarchical splitting of the modal DG basis, resulting in two weakly coupled problems for the low-order and high-order components of the solution (coarse and fine scale). While a global linear system of reduced size is solved for the coarse-scale problem, the fine-scale components are updated locally. IHSS extends the original hierarchical scale separation method, using an iterative solver to approximate the coarse-scale problem and shifting more work to the highly parallel local fine-scale updates. Convergence and computational performance of IHSS are evaluated using example problems from an application in the oil and gas industry, the simulation of the phase separation of binary fluid mixtures in three spatial dimensions. The problem is modeled by the Cahn–Hilliard equation, a fourth-order, nonlinear partial differential equation, which is discretized using the nonsymmetric interior penalty DG method. Numerical experiments demonstrate the applicability of IHSS to the linear systems arising in this problem. It is shown that their solution can be significantly accelerated when common iterative methods are used as coarse-scale solvers within IHSS instead of being applied directly. All parameters of the method are discussed in detail, and their impact on computational performance is evaluated.