GPU Optimizations for the Hierarchical Poincar\'e-Steklov Scheme

Abstract

This manuscript presents GPU optimizations for the 2D Hierarchical Poincar\'e-Steklov (HPS) discretization scheme. HPS is a multi-domain spectral collocation method that combines high-order discretizations with direct solvers to accurately resolve highly oscillatory solutions. The domain decomposition approach of HPS connects domains directly via a sparse direct solver. The proposed optimizations exploit batched linear algebra on modern hybrid architectures, are straightforward to implement, and improve the solver's practical speed. The manuscript demonstrates that GPU optimizations can significantly reduce the traditionally high cost of performing local static condensation for discretizations with very high local order $p$. Numerical experiments for the Helmholtz equation with high wavenumbers on curved and rectangular domains confirm the high accuracy achieved by the HPS discretization and the significant reduction in computation time achieved with GPU optimizations.