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The generalized Dryja--Smith--Widlund (GDSW) preconditioner is a two-leveloverlapping Schwarz domain decomposition (DD) preconditioner that couples aclassical one-level overlapping Schwarz preconditioner with anenergy-minimizing coarse space. When used to accelerate the convergence rate ofKrylov subspace iterative methods, the GDSW preconditioner provides robustnessand scalability for the solution of sparse linear systems arising from thediscretization of a wide range of partial different equations. In this paper,we present FROSch (Fast and Robust Schwarz), a domain decomposition solverpackage which implements GDSW-type preconditioners for both CPU and GPUclusters. To improve the solver performance on GPUs, we use a noveldecomposition to run multiple MPI processes on each GPU, reducing both solver'scomputational and storage costs and potentially improving the convergence rate.This allowed us to obtain competitive or faster performance using GPUs comparedto using CPUs alone. We demonstrate the performance of FROSch on the Summitsupercomputer with NVIDIA V100 GPUs, where we used NVIDIA Multi-Process Service(MPS) to implement our decomposition strategy.  The solver has a wide variety of algorithmic and implementation choices,which poses both opportunities and challenges for its GPU implementation. Weconduct a thorough experimental study with different solver options includingthe exact or inexact solution of the local overlapping subdomain problems on aGPU. We also discuss the effect of using the iterative variant of theincomplete LU factorization and sparse-triangular solve as the approximatelocal solver, and using lower precision for computing the whole FROSchpreconditioner. Overall, the solve time was reduced by factors of about$2\times$ using GPUs, while the GPU acceleration of the numerical setup timedepend on the solver options and the local matrix sizes.

An Experimental Study of Two-Level Schwarz Domain Decomposition Preconditioners on GPUs