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

B. Gmeiner, T. Gradl, F. Gaspar, U. Rüde
2013 Computers and Mathematics with Applications  
In this paper a local Fourier analysis for multigrid methods on tetrahedral grids is presented.  ...  Different smoothers for the discretization of the Laplace operator by linear finite elements on such grids are analyzed.  ...  in Bayern (KON-WIHR), and the International Doctorate Program (IDK) "Identification, Optimization and Control with Applications in Modern Technologies" within the Elite Network of Bavaria.  ... 
doi:10.1016/j.camwa.2012.12.006 fatcat:al7ipzs3nvb3zert3v6q7jw5hm

Multigrid fourier analysis on semi‐structured anisotropic meshes for vector problems

F.J. Gaspar, F.J. Lisbona, C. Rodrigo
2010 Mathematical Modelling and Analysis  
To this end, local Fourier analysis (LFA) on triangular grids for scalar problems is extended to the vector case.  ...  In order to choose the type of multigrid cycle and the number of pre-and post-smoothing steps, a three-grid Fourier analysis is done.  ...  Fourier Analysis General definitions Local Fourier Analysis is a powerful tool for the design of efficient multigrid methods on regular structured grids.  ... 
doi:10.3846/1392-6292.2010.15.39-54 fatcat:5psghb6ivfgd7cjrdqskls26om

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  ...  The work is motivated 425  ...  A popular explicit multigrid smoother is the semi-dis-relatively localized for transonic flows since the stiffness is only substantial near the stagnation point, at shocks, crete scheme proposed by Jameson  ... 
doi:10.1006/jcph.1997.5772 fatcat:yhpivovdazgrdgrrupdz7hn36y

hp-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part I. Multilevel analysis

J.J.W. van der Vegt, S. Rhebergen
2012 Journal of Computational Physics  
The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for low and high cell Reynolds numbers and on highly stretched meshes.  ...  The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge-Kutta method as smoother.  ...  The research of S.R. is partly funded by EU sixth framework ADIGMA project.  ... 
doi:10.1016/j.jcp.2012.05.038 fatcat:lgdyw62nenggvkoydpihuxm324

On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics

Francisco J. Gaspar, Carmen Rodrigo
2017 Computer Methods in Applied Mechanics and Engineering  
A local Fourier analysis is applied to study the convergence of the multigrid method and to support the good convergence results obtained.  ...  The proposed geometric multigrid algorithm is used to solve several tests in semi-structured triangular grids, in order to show the good behaviour of the method and its practical utility.  ...  Acknowledgements The work of Francisco J. Gaspar is supported by the European Union's  ... 
doi:10.1016/j.cma.2017.08.025 fatcat:7o6hxgvi3rdj5dhg2rps3e6nre

Local Fourier Analysis of P-Multigrid for High-Order Finite Element Operators [article]

Jeremy L Thompson, Jed Brown, Yunhui He
2021 arXiv   pre-print
Multigrid methods are popular for solving linear systems derived from discretizing PDEs. Local Fourier Analysis (LFA) is a technique for investigating and tuning multigrid methods.  ...  P-multigrid is popular for high-order or spectral finite element methods, especially on unstructured meshes.  ...  Local Fourier Analysis for P-Multigrid.  ... 
arXiv:2108.01751v1 fatcat:4i5nxcfabfbcjlab4lahbyswd4

Local Fourier Analysis of Multigrid Methods with Polynomial Smoothers and Aggressive coarsening [article]

James Brannick, Xiaozhe Hu, Carmen Rodrigo, Ludmil Zikatanov
2014 arXiv   pre-print
Using local Fourier analysis we determine automatically the optimal values for the parameters involved in defining the polynomial smoothers and achieve fast convergence of cycles with aggressive coarsening  ...  The methods we introduce are highly parallelizable and efficient multigrid algorithms on structured and semi-structured grids in two and three spatial dimensions.  ...  The work of Carmen Rodrigo is supported in part by the Spanish project FED-ER/MCYT MTM2010-16917 and the DGA (Grupo consolidado PDIE).  ... 
arXiv:1310.8385v2 fatcat:mgh4gkcxfjf7papetuka7d5odm

Fourier Analysis of GMRES(m) Preconditioned by Multigrid

Roman Wienands, Cornelis W. Oosterlee, Takumi Washio
2000 SIAM Journal on Scientific Computing  
Fourier analysis is a well-known useful tool in the multigrid community for the prediction of two-grid convergence rates ( 1], 8] ).  ...  This paper deals with convergence estimations of a preconditioned GMRES(m) method 6], where multigrid ( 1], 8]) is used as the preconditioner.  ...  Rigorous Fourier Analysis of Multigrid The 3D Poisson equation, a simple test problem, is chosen to explain the possibilities of convergence estimations by Fourier analysis for multigrid as a solver.  ... 
doi:10.1137/s1064827599353014 fatcat:55udjxbuc5gorkifwnyebwouxa

Multigrid Methods for Anisotropic Edge Refinement

Thomas Apel, Joachim Schöberl
2002 SIAM Journal on Numerical Analysis  
A nite element method with optimal convergence on non-smooth three dimensional domains requires anisotropic mesh re nement towards the edges.  ...  In this paper we suggest and analyze a new multigrid scheme combining semi-coarsening and line smoothers to obtain a solver of optimal algorithmic complexity for anisotropic meshes along edges.  ...  1 u l )k 0 ku l k A Dividing by one factor gives (18) and thus the desired approximation property. Theorem 6 (Convergence rate estimate).  ... 
doi:10.1137/s0036142900375414 fatcat:dbuvipz6wradjlrazsdslof2bi

h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

C.M. Klaij, M.H. van Raalte, H. van der Ven, J.J.W. van der Vegt
2007 Journal of Computational Physics  
Twolevel Fourier analysis of this method for the scalar advection-diffusion equation shows convergence factors between 0.5 and 0.75.  ...  The overall performance, therefore, highly depends on the efficiency of the solver.  ...  The work of Marc van Raalte has been supported by the Computational Life Science program of the Netherlands Organisation of Scientific Research (nwo 635.100.009) (C-pump project).  ... 
doi:10.1016/j.jcp.2007.08.034 fatcat:gg6dkywyjnhijn2g5kg2os7cd4

Using hierarchical matrices in the solution of the time-fractional heat equation by multigrid waveform relaxation [article]

Xiaozhe Hu, Carmen Rodrigo, Francisco J. Gaspar
2020 arXiv   pre-print
The efficiency and the good convergence of the algorithm, which can be theoretically justified by a semi-algebraic mode analysis, are demonstrated through numerical experiments in both one- and two-dimensional  ...  This work deals with the efficient numerical solution of the time-fractional heat equation discretized on non-uniform temporal meshes.  ...  Within these tests, a semi-algebraic mode analysis is used to theoretically justify the good convergence rates provided by the multigrid waveform relaxation algorithm.  ... 
arXiv:1706.07632v3 fatcat:y4pdytyisje3fo5qalihgfbl3i

On the robustness of ILU smoothers on triangular grids

M.A.V. Pinto, C. Rodrigo, F.J. Gaspar, C.W. Oosterlee
2016 Applied Numerical Mathematics  
A local Fourier analysis is proposed to study the smoothing properties of these methods, as well as the asymptotic convergence of the whole multigrid procedure.  ...  With this purpose, two-and three-grid local Fourier analysis are performed.  ...  The work of Francisco J. Gaspar and Carmen Rodrigo is supported in part by the Spanish project FEDER/MCYT MTM2013-40842-P and the DGA (Grupo consolidado PDIE).  ... 
doi:10.1016/j.apnum.2016.02.007 fatcat:72asoypbsve6zb6bwgt4bi2bke

A multigrid method for constrained optimal control problems

M. Engel, M. Griebel
2011 Journal of Computational and Applied Mathematics  
We present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying (elliptic) PDE problem.  ...  solution of linear-quadratic optimal control problems with additional constraints on the control.  ...  Acknowledgments The authors gratefully acknowledge the support from the Deutsche Forschungsgemeinschaft DFG through the Sonderforschungsbereich 611 ''Singular Phenomena and Scaling in Mathematical Models  ... 
doi:10.1016/j.cam.2011.04.002 fatcat:2g6utu646jcozfbrq5b2fpchkq

Multigrid and Runge-Kutta time stepping applied to the uniformly non-oscillatory scheme for conservation laws

J. W. van der Burg, J. G. M. Kuerten, P. J. Zandbergen
1991 Journal of Engineering Mathematics  
A Fourier analysis will be applied to investigate the optimal damping rate of all Fourier components of the error in the multigrid process.  ...  Finally, the optimal amplification rate in the numerical calculation is larger than the upper bound found with the Fourier analysis.  ... 
doi:10.1007/bf00044333 fatcat:7zqillluqbam7ihoitnowln7ea

Multigrid finite element methods on semi-structured triangular grids for planar elasticity

F. J. Gaspar, F. J. Lisbona, J. L. Gracia, C. Rodrigo
2009 Numerical Linear Algebra with Applications  
Acknowledgments This research has been partially supported by FEDER/MCYT Projects MTM2007-63204 and the DGA (Grupo consolidado PDIE).  ...  Multigrid solver on semi-structured grids As is well known, the construction of an efficient multigrid method is strongly dependent on the choice of its components.  ...  In the context of discretizations on semi-structured grids, particularly in the case of the hierarchical triangular meshes considered in this paper, an LFA is used to predict the behavior of the multigrid  ... 
doi:10.1002/nla.684 fatcat:lkr45anljrhs7kdezb72mwntxu
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