Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes

Brenner, Susanne C.; Cui, Jintao; Gudi, Thirupathi; Sung, Li-yeng

doi:10.1007/s00211-011-0379-y

Cited by 27 publications

(43 citation statements)

References 37 publications

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“…In 2003 Gopalakrishnan and Kanschat [6] analyzed a V-cycle preconditioner for diffusion and advectiondiffusion problems. Multigrid algorithms for dG discretizations of elliptic problems were considered by Brenner et al [7], who proved uniform convergence with respect to the number of levels for F-,V-and W-cycle on graded meshes, and Antonietti et al [8], who provided similar results for W-cycle h-,p-and hp-multigrid. While the previous works employed h-refined mesh sequences, Prill et al [9] considered smoothed aggregation to build coarse problems for h-multigrid dG solvers.…”

Section: Introductionmentioning

confidence: 98%

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

Botti

Colombo

Bassi

2017

Journal of Computational Physics

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In this work we exploit agglomeration based h-multigrid preconditioners to speed-up the iterative solution of discontinuous Galerkin discretizations of the Stokes and Navier-Stokes equations. As a distinctive feature h-coarsened mesh sequences are generated by recursive agglomeration of a fine grid, admitting arbitrarily unstructured grids of complex domains, and agglomeration based discontinuous Galerkin discretizations are employed to deal with agglomerated elements of coarse levels. Both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated. For the sake of efficiency coarse grid operators are inherited through element-by-element L 2 projections, avoiding the cost of numerical integration over agglomerated elements. Specific care is devoted to the projection of viscous terms discretized by means of the BR2 dG method. We demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels. The numerical solution of steady and unsteady, linear and non-linear problems is considered tackling challenging 2D test cases and 3D real life computations on parallel architectures.Significant execution time gains are documented.

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Section: Introductionmentioning

confidence: 98%

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

Botti

Colombo

Bassi

2017

Journal of Computational Physics

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“…In [40,41] a Fourier analysis of a multigrid solver for a class of DG discretizations is performed, focusing on the performance of several relaxation methods, while in [53] the analysis concerns convection-diffusion equations in the convection-dominated regime. Other contributions can be found for low-order DG approximations: in [22] it is shown that V-cycle, F-cycle and W-cycle multigrid algorithms converge uniformly with respect to all grid levels, with further extensions to an over-penalized method in [19] and graded meshes in [18,17]. At the best of our knowledge, no theoretical results in the framework of p-and hp-DG methods are available, even though p-multigrid solvers are widely used in practical applications, cf.…”

mentioning

confidence: 99%

Multigrid Algorithms for $hp$-Discontinuous Galerkin Discretizations of Elliptic Problems

Antonietti¹,

Sarti²,

Verani³

2015

SIAM J. Numer. Anal.

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We present W-cycle multigrid algorithms for the solution of the linear system of equations arising from a wide class of hp-version discontinuous Galerkin discretizations of elliptic problems. Starting from a classical framework in multigrid analysis, we define a smoothing and an approximation property, which are used to prove the uniform convergence of the W-cycle scheme with respect to the granularity of the grid and the number of levels. The dependence of the convergence rate on the polynomial approximation degree p is also tracked, showing that the contraction factor of the scheme deteriorates with increasing p. A discussion on the effects of employing inherited or non-inherited sublevel solvers is also presented. Numerical experiments confirm the theoretical results.

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“…This came after an additive multigrid convergence theory was developed in for conforming finite element methods that establishes the decay rate of the contraction numbers and completes the generalization of . The results in have since been extended to other nonconforming methods and DG methods .…”

Section: The Poisson Problemmentioning

confidence: 97%

Forty Years of the Crouzeix‐Raviart element

Brenner

2014

Numerical Methods Partial

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Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes

Cited by 27 publications

References 37 publications

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

Multigrid Algorithms for $hp$-Discontinuous Galerkin Discretizations of Elliptic Problems

Forty Years of the Crouzeix‐Raviart element

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