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Symbol Based Convergence Analysis in Block Multigrid Methods with applications for Stokes problems [article]

Matthias Bolten and Paola Ferrari and Isabella Furci
2021 arXiv   pre-print
In particular, we propose a symbol based convergence analysis for problems that have a hidden block Toeplitz structure.  ...  We show the efficiency of the methods studying the hidden 9× 9 block structure of the obtained matrix sequence proposing an efficient algebraic multigrid method with convergence rate independent of the  ...  In this paper we provided an efficient symbol-based multigrid method for block linear systems stemming from problems with a saddle-point structure.  ... 
arXiv:2107.07945v1 fatcat:65dq2fva2neh7gy6yuer5q5bru

Non-overlapping block smoothers for the Stokes equations [article]

Lisa Claus, Matthias Bolten
2020 arXiv   pre-print
However, the efficiency of multigrid methods for solving more difficult problems such as the Stokes equations lead to computationally more expensive smoothers, e.g., overlapping block smoothers.  ...  Traditional geometric multigrid methods are based on simple pointwise smoothers.  ...  Convergence behavior of the multigrid method for the Stokes system with periodic boundary conditions.  ... 
arXiv:2008.08719v1 fatcat:rr7bawuizfamjgdbt2xdthlg4a

Low-order preconditioning of the Stokes equations [article]

Alexey Voronin, Yunhui He, Scott MacLachlan, Luke N. Olson, Ray Tuminaro
2021 arXiv   pre-print
In particular, we investigate the use of geometric multigrid based on the ℚ_1isoℚ_2/ ℚ_1 discretization of the Stokes operator as a preconditioner for the ℚ_2/ℚ_1 discretization of the Stokes system.  ...  We utilize local Fourier analysis to optimize the damping parameters for Vanka and Braess-Sarazin relaxation schemes and to achieve robust convergence.  ...  ., for the U.S. Department of Energy's National Nuclear Security Administration under grant DE-NA-0003525. This paper describes objective technical results and analysis.  ... 
arXiv:2103.11967v2 fatcat:3lfhzbnwxjfwlawnl7cnkfs3xa

Nonoverlapping block smoothers for the Stokes equations

Lisa Claus, Matthias Bolten
2021 Numerical Linear Algebra with Applications  
However, using multigrid methods to efficiently solve more difficult problems such as the Stokes equations leads to computationally more expensive smoothers, for example, overlapping block smoothers.  ...  Traditional geometric multigrid methods are based on simple pointwise smoothers.  ...  ACKNOWLEDGEMENTS This work was partly supported by the German Research Foundation (DFG) through Priority Programme 1648, "Software for Exascale Computing" (SPPEXA).  ... 
doi:10.1002/nla.2389 fatcat:eozfxf2mbbdh3pees2rv3mzwlu

Efficient Variable-Coefficient Finite-Volume Stokes Solvers

Mingchao Cai, Andy Nonaka, John B. Bell, Boyce E. Griffith, Aleksandar Donev
2014 Communications in Computational Physics  
comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts.  ...  ., 228 (2009), pp. 7565-7595],as well; established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz  ...  As detailed in Appendix B, for constant-coefficient Stokes problems with periodic boundaries, GMRES will converge in a single iteration with preconditioner P 1 and in two iterations with P 2 and P 3 .  ... 
doi:10.4208/cicp.070114.170614a fatcat:jbb7zkn3fjhnzduxpnxfvpvqsu

Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

Peiyao Luo, Carmen Rodrigo, Francisco J. Gaspar, Cornelis W. Oosterlee
2017 SIAM Journal on Scientific Computing  
Other efficient multigrid methods for Stokes and incompressible Navier-Stokes problems have been developed, for example, in [5, 23, 28, 31, 46] .  ...  We deal with the so-called multiblock multigrid method which is based on the grid partitioning technique [45] .  ...  We thank Prashant Kumar for his help and discussion regarding the MATLAB code for the heterogeneous test.  ... 
doi:10.1137/16m1076514 fatcat:mbh3kic7lbdgjnckcy3aozxd44

Preconditioned Multigrid Methods for Compressible Flow Calculations on Stretched Meshes

Niles A. Pierce, Michael B. Giles
1997 Journal of Computational Physics  
On the other hand, the development of efficient numerical methods for solution of the Navier-Stokes equa-Efficient preconditioned multigrid methods are developed for both inviscid and viscous flow applications  ...  Convergence problems for Navier-Stokes applications the relaxation scheme on each mesh must successfully damp error modes.  ...  Numeri-tions with characteristic-based matrix dissipation is cal results for both the Euler and Navier-Stokes equations indicate that Fourier analysis does provide a useful indica- W t ϩ AW x ϭ ⌬x 2 ͉A͉W  ... 
doi:10.1006/jcph.1997.5772 fatcat:yhpivovdazgrdgrrupdz7hn36y

New multigrid smoothers for the Oseen problem

Steven Hamilton, Michele Benzi, Eldad Haber
2010 Numerical Linear Algebra with Applications  
Local Fourier analysis and numerical experiments on a 2-D lid-driven cavity problem indicate that the proposed smoothers result in hindependent convergence and are fairly robust with respect to the Reynolds  ...  We investigate the performance of smoothers based on the Hermitian/skew-Hermitian (HSS) and augmented Lagrangian (AL) splittings applied to MAC discretizations of the Oseen problem.  ...  We note in passing that for the Stokes problem (v = 0 in the Oseen problem) one can take ν = 1.  ... 
doi:10.1002/nla.707 fatcat:vufppcep55gqtgry3xetuhmsb4

A local Fourier analysis of additive Vanka relaxation for the Stokes equations [article]

Patrick E. Farrell, Yunhui He, Scott P. MacLachlan
2020 arXiv   pre-print
In this paper, we develop a local Fourier analysis of monolithic multigrid methods based on additive Vanka relaxation schemes for mixed finite-element discretizations of the Stokes equations.  ...  Multigrid methods are popular solution algorithms for many discretized PDEs, either as standalone iterative solvers or as preconditioners, due to their high efficiency.  ...  CONCLUSIONS AND FUTURE WORK We present a local Fourier analysis for a monolithic multigrid method based on overlapping additive Vanka-type relaxation for the Stokes equations.  ... 
arXiv:1908.09949v2 fatcat:uhn7leedlbe2bbmxuhujopyywu

Scalable robust solvers for unstructured FE geodynamic modeling applications: Solving the Stokes equation for models with large localized viscosity contrasts

T. Geenen, M. ur Rehman, S. P. MacLachlan, G. Segal, C. Vuik, A. P. van den Berg, W. Spakman
2009 Geochemistry Geophysics Geosystems  
1] The development of scalable robust solvers for unstructured finite element applications related to viscous flow problems in earth sciences is an active research area.  ...  We integrate AMG in our solver scheme as a preconditioner to the conjugate gradient method (CG) used during the construction of a block triangular preconditioner (BTR) to the Stokes equation, accelerating  ...  (On iterative methods for the incompressible Stokes problem, submitted to International Journal for Numerical Methods in Fluids, 2009).  ... 
doi:10.1029/2009gc002526 fatcat:pgl7fztnx5dc7o5rjz2p2kbhg4

On Bi-grid Local Mode Analysis of Solution Techniques for 3-D Euler and Navier–Stokes Equations

S.O. Ibraheem, A.O. Demuren
1996 Journal of Computational Physics  
ABSTRACT A procedure is presented for utilizing a bi-grid stability analysis as a practical tool for predicting multigrid performance in a range of numerical methods for solving Euler and Navier-Stokes  ...  Both predictions are compared with practical multigrid convergence rates for 2-D Euler and Navier-Stokes solutions based on the Beam-Warming central scheme.  ... 
doi:10.1006/jcph.1996.0099 fatcat:4jkyru2henacpect2jhiigdvsy

Monolithic multigrid for a reduced-quadrature discretization of poroelasticity [article]

James H. Adler, Yunhui He, Xiaozhe Hu, Scott MacLachlan, Peter Ohm
2022 arXiv   pre-print
Local Fourier analysis is used to optimize parameters in the resulting monolithic multigrid method, allowing a fair comparison between the performance and costs of methods based on Vanka and Braess-Sarazin  ...  This paper was motivated by an attempt to develop monolithic multigrid preconditioners for the discretization developed in [48]; we show here why this is a difficult task and, as a result, we modify the  ...  in the multigrid method.  ... 
arXiv:2107.04060v3 fatcat:qhq5h3oigvhp5hpt3hrenfaz5q

Symbol based convergence analysis in multigrid methods for saddle point problems [article]

Matthias Bolten and Marco Donatelli and Paola Ferrari and Isabella Furci
2022 arXiv   pre-print
The analysis is based on the generating symbols of the circulant blocks.  ...  Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations.  ...  Multigrid methods The TGM is useful for practical and preliminary convergence analysis of a multigrid method, but in practical applications, even the coarser problem is too large to be solved directly.  ... 
arXiv:2203.05959v1 fatcat:awjzyrcodfddjecdtu4pbucn2e

ExaStencils: Advanced Multigrid Solver Generation [chapter]

Christian Lengauer, Sven Apel, Matthias Bolten, Shigeru Chiba, Ulrich Rüde, Jürgen Teich, Armin Größlinger, Frank Hannig, Harald Köstler, Lisa Claus, Alexander Grebhahn, Stefan Groth (+5 others)
2020 Lecture Notes in Computational Science and Engineering  
At every layer, the corresponding language expresses not only computational directives but also domain knowledge of the problem and platform to be leveraged for optimization.  ...  Project ExaStencils pursued a domain-specific approach with a language, called ExaSlang, that is stratified into four layers of abstraction, the most abstract being the formulation in continuous mathematics  ...  Acknowledgments Project ExaStencils received generous funding from the Deutsche Forschungsgemeinschaft (DFG) in its priority programme SPP 1648 "Software for Exascale Computing" under grant numbers AP  ... 
doi:10.1007/978-3-030-47956-5_14 fatcat:xhbxnt45ynhilgh6vv2ui2n76i

On an Uzawa smoother in multigrid for poroelasticity equations

P. Luo, C. Rodrigo, F. J. Gaspar, C. W. Oosterlee
2016 Numerical Linear Algebra with Applications  
Here, for the efficient solution of the poroelastic equations, a multigrid method is employed with an Uzawa-type iteration as the smoother. The Uzawa smoother is an equation-wise procedure.  ...  In order to improve the convergence performance, the acceleration of multigrid by iterant recombination is taken into account.  ...  Detailed research of this method for a family of Stokes problems has already been done in. 9 A suitable relaxation parameter is chosen by means of local Fourier analysis (LFA). 10 LFA is a powerful  ... 
doi:10.1002/nla.2074 fatcat:ceu7lg5h4bavzgqxs774v5abca
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