Interior penalty tensor-product preconditioners for high-order discontinuous Galerkin discretizations

Pazner, Will; Persson, Per‐Olof

doi:10.2514/6.2018-1093

Cited by 4 publications

(4 citation statements)

References 28 publications

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“…Finally, we note that in the all of the above examples, the observed time step threshold corresponds to a CFL number of about 1/20, which is about standard for a DG method using degree nine polynomials [53,44,18]. (In the presented convergence plots of temporal accuracy, e.g., Figs.…”

Section: Convergence and Order Reduction For Collocation Formulationsmentioning

confidence: 74%

Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Minion

Saye

2018

Journal of Computational Physics

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This paper compares and contrasts higher-order, semi-implicit temporal integration strategies for the incompressible Navier-Stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations. Particular focus is placed on the imposition of boundary conditions in the semi-implicit formulation, the accurate treatment of the pressure term, and the smoothness of the numerical solution for different formulations. The main result presented here is the formulation and numerical validation of a single-step, semi-implicit method called spectral deferred pressure corrections, which can in theory obtain arbitrary formal order of accuracy in time. Numerical results demonstrate up to eighth-order accuracy, scenarios where optimal temporal accuracy is attained, and scenarios where order reduction is observed due to time-dependent boundary conditions.

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Section: Convergence and Order Reduction For Collocation Formulationsmentioning

confidence: 74%

Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Minion

Saye

2018

Journal of Computational Physics

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“…Therefore, this projection method computes u n+1 in three steps. First, the extrapolated contributions from the nonlinear and forcing terms are combined to compute F * (u n ) via (42). Second, we solve for p n+1 in the pressure-Poisson problem (46), closed with (47) and (48).…”

Section: Projection Methodsmentioning

confidence: 99%

“…Matrix-free multigrid methods using point Jacobi and Chebyshev smoothing were considered in [45] and [35]. Matrix-free tensor-product approximations to block Jacobi preconditioners for discontinuous Galerkin discretizations were constructed in [41] and [42]. In this work, we extend sparse, low-order refined preconditioners [36,22,16] with parallel subspace corrections, originally described for diffusion problems in [39].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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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, as described for diffusion problems in [39]. 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.

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“…[1]) were considered in [37]. Matrix-free approximate block Jacobi methods using Kronecker product approximations were constructed for discontinuous Galerkin discretizations of conservation laws in [43] and extended to interior penalty discretizations in [44].…”

Section: Introductionmentioning

confidence: 99%

Efficient low-order refined preconditioners for high-order matrix-free continuous and discontinuous Galerkin methods

Pazner

2019

Preprint

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In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on FEM-SEM equivalence and additive Schwarz methods. The high-order operators are applied without forming the system matrix, making use of sum factorization for efficient evaluation. The system is preconditioned using a spectrally equivalent low-order finite element operator discretization on a refined mesh. The low-order refined mesh is anisotropic and not shape regular in p, requiring specialized solvers to treat the anisotropy. We make use of a structured, geometric multigrid V-cycle with ordered ILU(0) smoothing. The preconditioner is parallelized through an overlapping additive Schwarz method that is robust in h and p. The method is extended to interior penalty and BR2 discontinuous Galerkin discretizations, for which it is also robust in the size of the penalty parameter. Numerical results are presented on a variety of examples, verifying the uniformity of the preconditioner.

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Interior penalty tensor-product preconditioners for high-order discontinuous Galerkin discretizations

Cited by 4 publications

References 28 publications

Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

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

Efficient low-order refined preconditioners for high-order matrix-free continuous and discontinuous Galerkin methods

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