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Local Fourier Analysis for Edge-Based Discretizations on Triangular Grids

Published online by Cambridge University Press:  03 March 2015

Carmen Rodrigo ,
Francisco Sanz ,
Francisco J. Gaspar  and
Francisco J. Lisbona
Carmen Rodrigo*
Affiliation:
Department of Applied Mathematics, University of Zaragoza, C/ Maria de Luna 3, 50018, Zaragoza, Spain
Francisco Sanz
Affiliation:
BIFI University of Zaragoza, C/ Pedro Cerbuna 12, 50009, Zaragoza, Spain
Francisco J. Gaspar
Affiliation:
Department of Applied Mathematics, University of Zaragoza, C/ Pedro Cerbuna 12, 50009, Zaragoza, Spain
Francisco J. Lisbona
Affiliation:
Department of Applied Mathematics, University of Zaragoza, C/ Pedro Cerbuna 12, 50009, Zaragoza, Spain
*
*Email addresses: carmenr@unizar.es (C. Rodrigo), frasanz@bifi.es (F. Sanz), fjgaspar@unizar.es (F. J. Gaspar), lisbona@unizar.es (F. J. Lisbona)
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Abstract

In this paper, we present a local Fourier analysis framework for analyzing the different components within multigrid solvers for edge-based discretizations on triangular grids. The different stencils associated with edges of different orientation in a triangular mesh make this analysis special. The resulting tool is demonstrated for the vector Laplace problem discretized by mimetic finite difference schemes. Results from the local Fourier analysis, as well as experimentally obtained results, are presented to validate the proposed analysis.

Keywords

65F1065N2265N55Multigridlocal Fourier analysisedge-based discretizationsmimetic finite differences
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 1 , February 2015 , pp. 78 - 96
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Arnold, D.N., Falk, R.S. AND Winther, R., Preconditioning in H(div) and applications, Math. Comp., 66 (1997), pp. 957–984.Google Scholar
[2]Beck, R., Algebraic multigrid by components splitting for edge elements on simplicial triangulations, Preprint SC 99–40, ZIB, Dec. 1999.Google Scholar
[3]Borzì, A., High-order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems, J. Comput. Appl. Math., 200 (2007), pp. 67–85.Google Scholar
[4]Bossavit, A., Computational Electromagnetism, Academic Press, San Diego, 1998.Google Scholar
[5]Brandt, A., Multi-level adaptive solutions to boundary-value problems, Math. Comput., 31 (1977), pp. 333–390.Google Scholar
[6]Brandt, A., Rigourous quantitative analysis of multigrid I. Constant coefficients two level cycle with L2 norm, SIAM J. Numer. Anal., 31 (1994), pp. 1695–1730.Google Scholar
[7]Gaspar, F.J., Gracia, J.L, Lisbona, F.J., Fourier Analysis for multigrid methods on triangular grids, SIAM J. Sci. Comput., 31 (2009), pp. 2081–2102.Google Scholar
[8]Hemker, P.W., Hoffmann, W. AND Van RAALTE, M.H., Fourier two-level analysis for discontinuous Galerkin discretization with linear elements, Numer. Linear Alg. Appl., 11 (2004), pp. 473–491.Google Scholar
[9]Hackbusch, W., Multi-grid methods and applications, Springer, Berlin, 1985.CrossRefGoogle Scholar
[10]Hiptmair, R. AND Xu, J., Nodal auxiliary space preconditioning in H(curl) and H(div) spaces, SIAM Journal Numerical Analysis, 45 (2007), pp. 2483–2509.Google Scholar
[11]Hu, J.J., Tuminaro, R.S., Bochev, P.B., Garasi, C.J., AND Robinson, A.C., Towards and h-independent algebraic multigrid method for Maxwell’s equations, SIAM Journal on Scientific Computing, 27 (2006), pp. 1669–1688.Google Scholar
[12]Lipnikov, K., Manzini, G., Shashkov, M., Mimetic finite difference methods, Journal of Computational Physics, 257 (2014), pp. 1163–1227.Google Scholar
[13]Reitzinger, S. AND Schöberl, J., An algebraic multigrid method for finite element discretizations with edge elements, Numer. Linear Alg. Appl., 9 (2002), pp. 223–238.Google Scholar
[14]Rodrigo, C., Gaspar, F.J. AND Lisbona, F.J., Geometric multigrid methods on triangular grids: Application to semi-structured meshes, Lambert Academic Publishing, Saarbrücken, 2012.Google Scholar
[15]Trottenberg, U., Oosterlee, C.W., Schüller, A., Multigrid, Academic Press, New York, 2001.Google Scholar
[16]Vabishchevich, P.N., Finite-difference approximation of mathematical physics problems on irregular grids, CMAM, 5 (2005), pp. 294–330.Google Scholar
[17]Weiland, T., A discretization method for the solution of Maxwell's equations for six-component fields, Electron. Commun. AEU 31 (1977), pp. 116–120.Google Scholar
[18]Wesseling, P., An Introduction to Multigrid Methods, John Wiley, Chichester, UK, 1992.Google Scholar
[19]Wienands, R., Joppich, W., Practical Fourier analysis for multigrid methods, Chapman and Hall/CRC Press, 2005.Google Scholar
[20]Wienands, R., Oosterlee, C.W. AND Washio, T.. Fourier analysis of GMRES(m) preconditioned by multigrid, SIAM J. Sci. Comput., 22 (2000), pp. 582–603.Google Scholar