Robust Multigrid for Cartesian Interior Penalty DG Formulations of the Poisson Equation in 3D

Stiller, Joerg

doi:10.1007/978-3-319-65870-4_12

Cited by 9 publications

(16 citation statements)

References 21 publications

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“…The pressure equations (71,76,81) and (85) are solved by means of a Krylov-accelerated polynomial multigrid technique using an elementbased overlapping Schwarz method for smoothing [82,83]. To cope with variable coefficients in diffusion problems (74,84) the Schwarz smoother was extended by adopting the linearization strategy developed in [81] for continuous spectral elements.…”

Section: Solution Methods and Implementationmentioning

confidence: 99%

A spectral deferred correction method for incompressible flow with variable viscosity

Stiller

2020

Preprint

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This paper presents a semi-implicit spectral deferred correction (SDC) method for incompressible Navier-Stokes problems with variable viscosity and time-dependent boundary conditions. The proposed method integrates elements of velocity-and pressure-correction schemes, which yields a simpler pressure handling in comparison to the SDPC method of Minion & Saye (J. Comput. Phys. 375: 797-822, 2018). Combined with the discontinuous Galerkin spectral-element method for spatial discretization it can in theory reach arbitrary order of accuracy in time and space. Numerical experiments in three space dimensions demonstrate up to order 12 in time and 17 in space for constant, spatiotemporally varying as well as solutiondependent viscosity. The phenomenon of order reduction also reported by Minion & Saye is observed in the case of time-dependent boundary conditions, where it manifests in terms of a slower convergence of the correction sweeps.

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Section: Solution Methods and Implementationmentioning

confidence: 99%

A spectral deferred correction method for incompressible flow with variable viscosity

Stiller

2020

Preprint

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“…In simulations the resolution often needs to be adapted to the solution, leading to anisotropic or even stretched meshes. To evaluate the effect of anisotropic meshes on the multigrid solvers the tests from [42] were repeated: The aspect ratio AR of the mesh varies from AR = 1 to AR = 48, expanding the domain to Ω = (0, 2π · AR) × (0, π AR/2 ) × (0, 2π) .…”

Section: Solver Runtimes For Anisotropic Meshesmentioning

confidence: 99%

“…While the solvers are very applicable for homogeneous meshes, their runtime increases rapidly for high-aspect ratios. Increasing the number of pre-and post-smoothing cycles per level slightly mitigates the problem and keeps the number of iterations mostly stable until AR = 8, but for higher aspect ratios a higher overlap is required [42].…”

Section: Solver Runtimes For Anisotropic Meshesmentioning

confidence: 99%

Scaling to the stars – a linearly scaling elliptic solver for p-multigrid

Huismann

Stiller

Fröhlich

2019

Journal of Computational Physics

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High-order methods gain increased attention in computational fluid dynamics. However, due to the time step restrictions arising from the semi-implicit time stepping for the incompressible case, the potential advantage of these methods depends critically on efficient elliptic solvers. Due to the operation counts of operators scaling with with the polynomial degree p times the number of degrees of freedom nDOF, the runtime of the best available multigrid solvers scales with O(p · nDOF). This scaling with p significantly lowers the applicability of high-order methods to high orders. While the operators for residual evaluation can be linearized when using static condensation, Schwarz-type smoothers require their inverses on fixed subdomains. No explicit inverse is known in the condensed case and matrix-matrix multiplications scale with p · nDOF. This paper derives a matrix-free explicit inverse for the static condensed operator in a cuboidal subdomain. It scales with p 3 per element, i.e. nDOF globally, and allows for a linearly scaling additive Schwarz smoother, yielding a p-multigrid cycle with an operation count of O(nDOF). The resulting solver uses fewer than four iterations for all polynomial degrees to reduce the residual by ten orders and has a runtime scaling linearly with nDOF for polynomial degrees at least up to 48. Furthermore the runtime is less than one microsecond per unknown over wide parameter ranges when using one core of a CPU, leading to time-stepping for the incompressible Navier-Stokes equations using as much time for explicitly treated convection terms as for the elliptic solvers. * Corresponding author: Immo.Huismann@tu-dresden.de arXiv:1808.03595v1 [math.NA] 10 Aug 2018 to h-multigrid. In both cases, smoothers are required and overlapping Schwarz smoothers are a good option [18,29,41]. These decompose the grid into overlapping blocks on which the exact inverse is applied, producing exceptional smoothing rates. But these methods not only require the super-linearly scaling operator for residual evaluation, but a super-linearly scaling solver as well. This super-linear scaling prevents the respective multigrid techniques from achieving their full potential and prohibits the usage of large polynomial degrees.The goal of this paper is to devise a linearly scaling multigrid cycle for three-dimensional structured Cartesian grids, where residual evaluation, smoothing, restriction, and prolongation all scale linearly with the number degrees of freedom, without a further factor of p present in other methods. To this end the residual evaluation derived in [22] for the static condensed case is combined with the p-multigrid block smoother technique proposed in [18,17], leaving only the inverse on a 2 3 element block as non-matrix-free, super-linearly scaling operator. This operator is embedded in the full system and then factorized and rearranged, attaining a matrix-free inverse that is then factorized to linear complexity. The inverse leads to a linearly scaling multigrid cycle, which is confirmed in run...

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“…The implicit discretization of diffusive terms leads to symmetric linear systems, for which very efficient solution methods exist, e.g. [23,34,51,52]. This offers an enormous advantage over fully implicit methods, especially in high fidelity simulations, where the time step is limited by accuracy rather than stability constraints.…”

Section: Introductionmentioning

confidence: 99%

“…The method was implemented in the HiSPEET 1 library which makes use of MPI for parallelization and LIBXSMM [22] for vectorizing the element operators. It also provides highly efficient Krylov-accelerated Schwarz/multigrid methods, which are used for solving the implicit pressure and diffusion problems [50][51][52].…”

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confidence: 99%

Assessment of high-order IMEX methods for incompressible flow

Montadhar¹,

Grotteschi²,

Stiller³

2021

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This paper investigates the competitiveness of semi-implicit Runge-Kutta (RK) and spectral deferred correction (SDC) time-integration methods up to order six for incompressible Navier-Stokes problems in conjunction with a high-order discontinuous Galerkin method for space discretization. It is proposed to harness the implicit and explicit RK parts as a partitioned scheme, which provides a natural basis for the underlying projection scheme and yields a straight-forward approach for accommodating nonlinear viscosity. Numerical experiments on laminar flow, variable viscosity and transition to turbulence are carried out to assess accuracy, convergence and computational efficiency. Although the methods of order 3 or higher are susceptible to order reduction due to time-dependent boundary conditions, two thirdorder RK methods are identified that perform well in all test cases and clearly surpass all second-order schemes including the popular extrapolated backward difference method. The considered SDC methods are more accurate than the RK methods, but become competitive only for relative errors smaller than ca 10 −5 .

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Robust Multigrid for Cartesian Interior Penalty DG Formulations of the Poisson Equation in 3D

Cited by 9 publications

References 21 publications

A spectral deferred correction method for incompressible flow with variable viscosity

A spectral deferred correction method for incompressible flow with variable viscosity

Scaling to the stars – a linearly scaling elliptic solver for p-multigrid

Assessment of high-order IMEX methods for incompressible flow

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