Multigrid method for anisotropic diffusion equations based on adaptive Chebyshev smoothers

Zhukov, V. T.; Novikova, N. D.; Феодоритова, О. Б.

doi:10.1134/s2070048215020118

Cited by 6 publications

(4 citation statements)

References 3 publications

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“…The projection of the discrepancy r h = g h − A h u h onto the H-grid by the operator R is taken as the right-hand side of the system on the H-grid. Algorithmic constructions of inter-grid transition operators P and R = P * are indicated in [19,20], where, in addition to the trilinear interpolation operator, the interpolation operator P based on the approximate solution of local boundary value problems is also given; this is critical for the case of discontinuous coefficients. Such inter-grid transition operators P and R = P * are said to be called problem-dependent (see [14]).…”

Section: Multigrid Methodsmentioning

confidence: 99%

“…We consider only the classical version of the multigrid method with resampling on all additional grid levels. Numerous meaningful examples of solving problems with various degrees of complexity (with discontinuous anisotropic coefficients and significant anisotropy) are given in [18][19][20][21]. For demonstration of the computational efficiency, we provide the results of the solution of the simplest equation u t = Δu + f in the parallelepiped…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…The amplitude of such components decreases almost inversely proportional to the frequency, which corresponds to the needs of the multigrid method. As a result, for the geometric classical multigrid method (MM) on Cartesian grids, we have constructed an effective parallel adaptive implementation (see [18][19][20][21]); now we develop it for the algebraic multigrid method.…”

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confidence: 99%

See 2 more Smart Citations

On Development of Parallel Algorithms for Solving Parabolic and Elliptic Equations

Zhukov

Феодоритова

2021

J Math Sci

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In this paper, we present results of the development of certain parallel numerical methods for solving three-dimensional evolutionary and stationary problems of diffusion and heat transfer. We present a detailed description of a special, explicit iteration scheme for parabolic equations and discuss a multigrid technology used for solving elliptic equations and implicit schemes for parabolic equations.

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Section: Multigrid Methodsmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

mentioning

confidence: 99%

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On Development of Parallel Algorithms for Solving Parabolic and Elliptic Equations

Zhukov

Феодоритова

2021

J Math Sci

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“…On the other hand, results are less sensitive to the minimum eigenvalue estimate, provided λ min > 0, as this delimits between the low frequencies that are handled by the coarse grid correction and the high frequencies that are eliminated by the action of the smoother. The choice of the bounds (relative to λ) is not universal: (1/30,1.1) [1], (0.3,1) [2], (0.25,1) [28], and (1/6,1) [32] have all been considered. Based on the results of Table 1, we set (λ min , λ max ) = (0.1, 1.1) λ as a conservative choice.…”

Section: Chebyshev Accelerationmentioning

confidence: 99%

Tuning Spectral Element Preconditioners for Parallel Scalability on GPUs

Phillips¹,

Kerkemeier²,

Fischer³

2022

Proceedings of the 2022 SIAM Conference on Parallel Processing for Scientific Computing (PP)

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The Poisson pressure solve resulting from the spectral element discretization of the incompressible Navier-Stokes equation requires fast, robust, and scalable preconditioning. In the current work, a parallel scaling study of Chebyshevaccelerated Schwarz and Jacobi preconditioning schemes is presented, with special focus on GPU architectures, such as OLCF's Summit. Convergence properties of the Chebyshevaccelerated schemes are compared with alternative methods, such as low-order preconditioners combined with algebraic multigrid. Performance and scalability results are presented for a variety of preconditioner and solver settings. The authors demonstrate that Chebyshev-accelerated-Schwarz methods provide a robust and effective smoothing strategy when using p-multigrid as a preconditioner in a Krylovsubspace projector. The variety of cases to be addressed, on a wide range of processor counts, suggests that performance can be enhanced by automated run-time selection of the preconditioner and associated parameters.

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An adaptive multigrid on block-structured grids

Феодоритова

Zhukov

2020

J. Phys.: Conf. Ser.

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An implementation of the multigrid method on conformal block-structured grids is proposed. The algorithm is intended to solve a boundary-value problem for elliptic PDEs. Each block is discretized using a structured hexahedron grid.The set of multilevel grids are arranged in hierarchical levels. In each block the discrete finite volume scheme is constructed on the fine level grid. The coarse level equations are formed by the Galerkin procedure. The presence of the irregular block connections leads to irregular stencils in the nodes which adjoin of several blocks. Instead of a special irregular node treatment we propose to find the solution in these nodes in additive manner. Such an opportunity is provided by the use of the explicit iterative procedures for both the smoothers and the coarsest grid solver. We also present an adaptive technique which adjusts these procedures for achieving the prescribed multigrid convergence rate. The proposed approach enhances the potential of the multigrid method in ultra-parallel computing.

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Multigrid method for anisotropic diffusion equations based on adaptive Chebyshev smoothers

Cited by 6 publications

References 3 publications

On Development of Parallel Algorithms for Solving Parabolic and Elliptic Equations

On Development of Parallel Algorithms for Solving Parabolic and Elliptic Equations

Tuning Spectral Element Preconditioners for Parallel Scalability on GPUs

An adaptive multigrid on block-structured grids

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