High-order matrix-free incompressible flow solvers with GPU acceleration and low-order refined preconditioners

TL;DR

This paper introduces a GPU-accelerated, matrix-free high-order finite element flow solver for incompressible Navier-Stokes and Stokes equations, utilizing low-order refined preconditioners for efficient linear system solutions.

Contribution

It develops novel matrix-free operators and preconditioners for high-order discretizations, optimized for GPU acceleration, improving computational efficiency and scalability.

Findings

Achieves efficient GPU-based matrix-free operator evaluations.

Develops robust preconditioners for saddle-point systems.

Demonstrates high performance on benchmark problems in 2D and 3D.

Abstract

We present a matrix-free flow solver for high-order finite element discretizations of the incompressible Navier-Stokes and Stokes equations with GPU acceleration. For high polynomial degrees, assembling the matrix for the linear systems resulting from the finite element discretization can be prohibitively expensive, both in terms of computational complexity and memory. For this reason, it is necessary to develop matrix-free operators and preconditioners, which can be used to efficiently solve these linear systems without access to the matrix entries themselves. The matrix-free operator evaluations utilize GPU-accelerated sum-factorization techniques to minimize memory movement and maximize throughput. The preconditioners developed in this work are based on a low-order refined methodology with parallel subspace corrections. The saddle-point Stokes system is solved using block-preconditioning techniques, which are robust in mesh size, polynomial degree, time step, and viscosity. For the incompressible Navier-Stokes equations, we make use of projection (fractional step) methods, which require Helmholtz and Poisson solves at each time step. The performance of our flow solvers is assessed on several benchmark problems in two and three spatial dimensions.