Inexact hierarchical scale separation: A two-scale approach for linear systems from discontinuous Galerkin discretizations

Abstract

Hierarchical scale separation (HSS) is an iterative two-scale approximation method for large sparse systems of linear equations arising from discontinuous Galerkin (DG) discretizations. HSS splits the linear system into a coarse-scale system of reduced size corresponding to the local mean values of the solution, and a set of decoupled local fine-scale systems corresponding to the higher-order solution components. This scheme then alternates between coarse-scale and fine-scale system solves until both components converge. The motivation of HSS is to promote parallelism by decoupling the fine-scale systems, and to reduce the communication overhead from classical linear solvers by only applying them to the coarse-scale system.We propose a modified HSS scheme (“inexact HSS”, “IHSS”) that exploits the highly parallel fine-scale solver more extensively and only approximates the coarse-scale solution in every iteration, thus resulting in a significant speedup. The tolerance of the coarse-scale solver is adapted in every IHSS cycle, controlled by the residual norm of the fine-scale system. Anderson acceleration is employed in the repeated solving of the fine-scale system to stabilize the scheme. We investigate the applicability of IHSS to systems stemming from the nonsymmetric interior penalty DG discretization of the Cahn–Hilliard equation, discuss its hybrid parallel implementation for large-scale simulations, and compare the performance of a widely used iterative solver with and without IHSS.