Inexact Hierarchical Scale Separation: An Efficient Linear Solver for Discontinuous Galerkin Discretizations

Abstract

Abstract Hierarchical scale separation (HSS) is a new approach to solve large sparse systems of linear equations arising from discontinuous Galerkin (DG) discretizations. We investigate its applicability 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 its performance to a widely used iterative solver. The solution of the linear systems, in particular in massively parallel applications, is a known performance bottleneck in direct numerical approaches. HSS splits the linear system into a coarse-scale system of reduced size corresponding to the local mean values of the DG solution, and a set of decoupled local fine-scale systems corresponding to the higher order components of the DG solution. The scheme then alternates between coarse-scale and fine-scale system solves until both components converge, employing a standard iterative solver for the coarse-scale system and direct solves for the set of small fine-scale systems, which allow for an optimal parallelization. The motivation of HSS is to increase parallelism by decoupling the linear systems, therefore reducing the communication overhead within sparse matrix-vector multiplications of classical iterative solvers. Providing some mild assumptions on the underlying DG basis functions, the above-mentioned splitting can be done on the resulting linear systems (i. e. without knowledge of the numerical scheme), which further motivates the development of the HSS scheme as a blackbox solver for DG discretizations. We propose a modified HSS algorithm ("inexact HSS," IHSS") that shifts computation to the highly parallel fine-scale solver, and thus reduces global synchronization. The key result is that the IHSS scheme significantly speeds up the linear system solves and outperforms a standard GMRES solver (up to 9x speedup for some configurations). A hybrid parallel IHSS solver has been implemented using the Trilinos package. Its convergence for linear systems from the Cahn-Hilliard problem is verified, and its performance is compared to a standard iterative solver from the same package. In the future, IHSS may possibly be used as a blackbox solver to speed up DG based simulations, e.g., of reservoir flow or multicomponent transport.