Optimization of the multigrid-convergence rate on semi-structured meshes by local Fourier analysis

Gmeiner, Björn; Gradl, T.; Gaspar, Francisco José; Rüde, Ulrich

doi:10.1016/j.camwa.2012.12.006

Cited by 21 publications

(16 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This is caused by tetrahedra with suboptimal shape. To recover the ideal convergence rates, modified smoothers could be used; see [25]. Table 2 presents weak scaling experiments on all three geometries, where the number of threads is increased by a factor of eight in each row.…”

Section: Comparison Of the Weak Scalability For The Different Geometrmentioning

confidence: 99%

Performance and Scalability of Hierarchical Hybrid Multigrid Solvers for Stokes Systems

Gmeiner¹,

Rüde²,

Stengel³

et al. 2015

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

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

show abstract

Section: Comparison Of the Weak Scalability For The Different Geometrmentioning

confidence: 99%

Performance and Scalability of Hierarchical Hybrid Multigrid Solvers for Stokes Systems

Gmeiner¹,

Rüde²,

Stengel³

et al. 2015

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…5. This can have negative impact on the convergence rate of a standard multigrid method, but this can be circumvented by appropriate line and plane relaxations in the block structured grid, see [27].…”

Section: Application To Flow Problemsmentioning

confidence: 99%

A quantitative performance study for Stokes solvers at the extreme scale

Gmeiner

Huber

John

et al. 2016

Journal of Computational Science

View full text Add to dashboard Cite

“…Local Fourier analysis has been traditionally performed for finite difference discretizations on structured rectangular grids. This analysis was extended to FE discretizations on general structured triangular [37,38] and tetrahedral [39] grids. The key fact for this extension is to consider an expression of the Fourier transform in new coordinate systems in space and frequency variables and introduce a non-orthogonal unit basis of R d , chosen to fit the geometry of the given simplicial mesh.…”

Section: Local Fourier Analysismentioning

confidence: 99%

A finite element framework for some mimetic finite difference discretizations

Rodrigo

Gaspar

et al. 2015

Computers & Mathematics with Applications

View full text Add to dashboard Cite

In this work we derive equivalence relations between mimetic finite difference schemes on simplicial grids and modified Nédélec-Raviart-Thomas finite element methods for model problems in H(curl) and H(div). This provides a simple and transparent way to analyze such mimetic finite difference discretizations using the well-known results from finite element theory. The finite element framework that we develop is also crucial for the design of efficient multigrid methods for mimetic finite difference discretizations, since it allows us to use canonical inter-grid transfer operators arising from the finite element framework. We provide special Local Fourier Analysis and numerical results to demonstrate the efficiency of such multigrid methods.

show abstract

Optimization of the multigrid-convergence rate on semi-structured meshes by local Fourier analysis

Cited by 21 publications

References 31 publications

Performance and Scalability of Hierarchical Hybrid Multigrid Solvers for Stokes Systems

Performance and Scalability of Hierarchical Hybrid Multigrid Solvers for Stokes Systems

A quantitative performance study for Stokes solvers at the extreme scale

A finite element framework for some mimetic finite difference discretizations

Contact Info

Product

Resources

About