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

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.

arXiv comment20 pages, 11 figures

References in corpus (6)

• Nodal Discontinuous Galerkin Methods on Graphics Processors
• Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method
• Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods
• Matrix-free multigrid block-preconditioners for higher order Discontinuous Galerkin discretisations
• A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows
• Short note on the mass matrix for Gauss-Lobatto grid points

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