Multigrid Algorithms for High Order Discontinuous Galerkin Methods

Antonietti, Paola F.; Sarti, Marco; Verani, Marco

doi:10.1007/978-3-319-18827-0_1

Cited by 8 publications

(9 citation statements)

References 24 publications

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“…To assess the impact these jumps have on the convergence of Parareal, Figure 5 gives a comparison of the defect for the brick and mortar problem (red) and a reference configuration with D lip = D cor = 10 −3 throughout the whole domain. For the setup studied here, in line with the findings for 2D problems in [31], the jump in coefficients has almost no effect on how Parareal convergence. Experiments not documented here suggest that a larger T (that is, a final configuration closer to the steady state) can lead to a larger detrimental effect of coefficient jumps: However, even there this only resulted in a small number of additional iterations required for convergence.…”

Section: Effect Of Spatially Varying Coefficientssupporting

confidence: 87%

“…problem, it already has jumps in the diffusion coefficients of several orders of magnitude on a highly anisotropic domain. The article is an extension of a previous study of a 2D problem on a domain with a much simpler structure [31].…”

Section: Discussionmentioning

confidence: 99%

“…Theory for diffusive problems with constant coefficients can also be found in [6]. For 2D diffusion with space-time dependent coefficients, numerical experiments showed only a marginal reduction in convergence speed [31]. In [4], the small impact of a time dependent viscosity on Parareal's convergence for an advection-diffusion problem is demonstrated.…”

Section: Introductionmentioning

confidence: 98%

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Numerical simulation of skin transport using Parareal

Kreienbuehl

Naegel

Ruprecht

et al. 2015

Comput. Visual Sci.

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In-silico investigation of skin permeation is an important but also computationally demanding problem. To resolve all scales involved in full detail will not only require exascale computing capacities but also suitable parallel algorithms. This article investigates the applicability of the time-parallel Parareal algorithm to a brick and mortar setup, a precursory problem to skin permeation. The C++ library Lib4PrM implementing Parareal is combined with the UG4 simulation framework, which provides the spatial discretization and parallelization. The combination's performance is studied with respect to convergence and speedup. It is confirmed that anisotropies in the domain and jumps in diffusion coefficients only have a minor impact on Parareal's convergence. The influence of load imbalances in time due to differences in number of iterations required by

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Section: Effect Of Spatially Varying Coefficientssupporting

confidence: 87%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Numerical simulation of skin transport using Parareal

Kreienbuehl

Naegel

Ruprecht

et al. 2015

Comput. Visual Sci.

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“…In this paper we analyze the convergence of a two-level scheme and W-cycle multigrid method for the solution of the linear system of equations arising from the hp-version of the interior penalty DG scheme on polygonal/polyhedral meshes [35], thereby, extending the theoretical framework developed in [12] for standard quasi-uniform triangular/quadrilateral meshes, cf. also [13] for three-dimensional numerical experiments. Our analysis is based on the smoothing and approximation properties associated with the proposed method: the former corresponds to a Richardson iteration, whose study requires a result concerning the spectral properties of the stiffness matrix, while the latter is inherent to the interior penalty DG scheme itself and exploits the error estimates derived in [35].…”

mentioning

confidence: 99%

Multigrid algorithms for $$\varvec{hp}$$ h p -version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes

Antonietti

Houston

et al. 2017

Calcolo

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In this paper we analyze the convergence properties of two-level and W-cycle multigrid solvers for the numerical solution of the linear system of equations arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polygonal/polyhedral meshes. We prove that the two-level method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. An analogous result is obtained for the W-cycle multigrid algorithm, which is proved to be uniformly convergent with respect to the mesh size, the polynomial approximation degree, and the number of levels, provided the number of smoothing steps is chosen sufficiently large. Numerical experiments are presented which underpin the theoretical predictions; moreover, the proposed Paola F. Antonietti has been partially supported by SIR (Scientific Independence of young Researchers) starting grant n. RBSI14VT0S "PolyPDEs: Non-conforming polyhedral finite element methods for the approximation of partial differential equations" funded by the Italian Ministry

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“…A number of works are available on multigrid for DG methods. Interior penalty methods are the most commonly analyzed, for instance, see [12], [11], [16], [2], and [1]. Most of these works are theoretical, and while they are able to prove convergence, the numerical experiments show rates below what is typically expected from GMG in this model setting.…”

mentioning

confidence: 99%

Manycore Parallel Computing for a Hybridizable Discontinuous Galerkin Nested Multigrid Method

Fabien¹,

Knepley²,

Mills³

et al. 2019

SIAM J. Sci. Comput.

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We present a heterogeneous computing strategy for a hybridizable discontinuous Galerkin (HDG) geometric multigrid (GMG) solver. Parallel GMG solvers require a combination of coarse grain and fine grain parallelism is utilized to improve time to solution performance. In this work we focus on fine grain parallelism. We use Intel's second generation Xeon Phi (Knights Landing) to enable acceleration. The GMG method achieves ideal convergence rates of 0.2 or less, for high polynomial orders. A matrix free (assembly free) technique is exploited to save considerable memory usage and increase arithmetic intensity. HDG enables static condensation, and due to the discontinuous nature of the discretization, we developed a matrix-vector multiply routine that does not require any costly synchronizations or barriers. Our algorithm is able to attain 80% of peak bandwidth performance for higher order polynomials. This is possible due to the data locality inherent in the HDG method. Very high performance is realized for high order schemes, due to good arithmetic intensity, which declines as the order is reduced.

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Multigrid Algorithms for High Order Discontinuous Galerkin Methods

Cited by 8 publications

References 24 publications

Numerical simulation of skin transport using Parareal

Numerical simulation of skin transport using Parareal

Multigrid algorithms for $$\varvec{hp}$$ h p -version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes

Manycore Parallel Computing for a Hybridizable Discontinuous Galerkin Nested Multigrid Method

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