Multigrid Methods for the Stokes Equations using Distributive Gauss–Seidel Relaxations based on the Least Squares Commutator

Wang, Ming; Chen, Long

doi:10.1007/s10915-013-9684-1

Cited by 48 publications

(31 citation statements)

References 45 publications

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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (cf. [47])…”

Section: Scalability To Extreme Resolutionsmentioning

confidence: 99%

Performance and Scalability of Hierarchical Hybrid Multigrid Solvers for Stokes Systems

Gmeiner¹,

Rüde²,

Stengel³

et al. 2015

SIAM J. Sci. Comput.

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In many applications involving incompressible fluid flow, the Stokes system plays an important role. Complex flow problems may require extremely fine resolutions, easily resulting in saddle-point problems with more than a trillion (10 12 ) unknowns. Even on the most advanced supercomputers, the fast solution of such systems of equations is a highly nontrivial and challenging task. In this work we consider a realization of an iterative saddle-point solver which is based mathematically on the Schur-complement formulation of the pressure and algorithmically on the abstract concept of hierarchical hybrid grids. The design of our fast multigrid solver is guided by an innovative performance analysis for the computational kernels in combination with a quantification of the communication overhead. Excellent node performance and good scalability to almost a million parallel threads are demonstrated on different characteristic types of modern supercomputers.1. Introduction. Current leading edge supercomputers can provide performance in the order of several petaflop/s, enabling the development of increasingly complex and accurate computational models having unprecedented size. This is especially relevant in flow simulations that may exhibit many small scale features that must be resolved over large domains. As an example, the problem of earth mantle convection is posed on a thick spherical shell of approximately 3 000 km depth and 6 300 km radius, resulting in an overall volume of close to a trillion, that is, 10 12 km 3 . A high resolution then results automatically in huge algebraic systems.Although finite element (FE) methods are flexible enough to handle different local mesh-sizes, fully adaptive meshing techniques require dynamic data structures and a complex program control flow that incurs significant computational cost. Recent work on parallel adaptive FE techniques can be found, e.g., in [1,2,11,44]. In [10] it is shown that an adaptive parallel FE method can reach locally 1 km resolution for the mantle convection problem on a large scale supercomputer. Here we will demonstrate that such a resolution can even be reached globally.Higher order FE approaches can lead to a better accuracy with the same number of unknowns, but the linear systems are denser. This implies more computational work, more memory access cost, and also higher parallel communication cost, so

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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (cf. [47])…”

Section: Scalability To Extreme Resolutionsmentioning

confidence: 99%

Performance and Scalability of Hierarchical Hybrid Multigrid Solvers for Stokes Systems

Gmeiner¹,

Rüde²,

Stengel³

et al. 2015

SIAM J. Sci. Comput.

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“…В настоящей работе давление задавалось в вершине ячейки Ω 1,1 или же исполь-зовалось условие (13). Если сторона ячейки совпадает с границей области Ω, то в соответствующих точках вместо условий согласования в локальной СЛАУ выписы-ваются граничные условия: u m = g m , m = 1, 2.…”

Section: описание метода кннunclassified

“…Вследствие этого матрица A col неполного ранга. Чтобы потенциально с по-мощью этой матрицы можно было определить решение, необходимо включить в A col строку, соответствующую уравнению (13). Эта строка имеет следующий вид: {0, 0, 0, 0, 0, 0, 0, 0, 0, h, 0, 0}.…”

Section: предобуславливатели для метода кннunclassified

“…Расчету без применения алгоритма Крылова соответствует случай k = 0. В этой серии расчетов в пере-определенную СЛАУ (14) было включено уравнение (13). В процессе итераций по методу КНН абсолютная величина интеграла (3) падала со значения порядка 10 −3 до величины порядка 10 −12 − 10 −13 , то есть до величины порядка машинных ошибок округления при расчетах по Фортран-программе с двойной точностью.…”

Section: тестированиеunclassified

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On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals

Shapeev¹,

Vorozhtsov²

2017

Model. anal. inf. sist.

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“…Среди конечно-разностных алгоритмов решения уравнений Стокса одним из наиболее эффектив-ных является многосеточный метод [13][14][15][16]. В данной работе мы также предлагаем реализацию и произ-водим тестирование численной схемы для многосеточного метода решения уравнений Стокса с перемен-ной вязкостью в сферической системе координат.…”

Section: Introductionunclassified

Benchmark solutions for Stokes equations with variable viscosity in cylindrical and spherical coordinates for algorithm testing

Makeev¹,

Popov²,

Blinova³

2016

Naučno-teh. vestn. inf. tehnol. meh. opt.

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Multigrid Methods for the Stokes Equations using Distributive Gauss–Seidel Relaxations based on the Least Squares Commutator

Cited by 48 publications

References 45 publications

Performance and Scalability of Hierarchical Hybrid Multigrid Solvers for Stokes Systems

Performance and Scalability of Hierarchical Hybrid Multigrid Solvers for Stokes Systems

On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals

Benchmark solutions for Stokes equations with variable viscosity in cylindrical and spherical coordinates for algorithm testing

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