Hostname: page-component-7c8c6479df-fqc5m Total loading time: 0 Render date: 2024-03-26T20:20:05.068Z Has data issue: false hasContentIssue false

Analysis of a Two-Level Algorithm for HDG Methods for Diffusion Problems

Published online by Cambridge University Press:  17 May 2016

Binjie Li*
Affiliation:
School of Mathematics, Sichuan University, Chengdu 610064, China
Xiaoping Xie*
Affiliation:
School of Mathematics, Sichuan University, Chengdu 610064, China
Shiquan Zhang*
Affiliation:
School of Mathematics, Sichuan University, Chengdu 610064, China
*
*Corresponding author. Email addresses:, libinjiefem@yahoo.com(B. Li), xpxie@scu.edu.cn(X. Xie), shiquanzhang@scu.edu.cn(S. Zhang)
*Corresponding author. Email addresses:, libinjiefem@yahoo.com(B. Li), xpxie@scu.edu.cn(X. Xie), shiquanzhang@scu.edu.cn(S. Zhang)
*Corresponding author. Email addresses:, libinjiefem@yahoo.com(B. Li), xpxie@scu.edu.cn(X. Xie), shiquanzhang@scu.edu.cn(S. Zhang)
Get access

Abstract

This paper analyzes an abstract two-level algorithm for hybridizable discontinuous Galerkin (HDG) methods in a unified fashion. We use an extended version of the Xu-Zikatanov (X-Z) identity to derive a sharp estimate of the convergence rate of the algorithm, and show that the theoretical results also are applied to weak Galerkin (WG) methods. The main features of our analysis are twofold: one is that we only need the minimal regularity of the model problem; the other is that we do not require the triangulations to be quasi-uniform. Numerical experiments are provided to confirm the theoretical results.

Type
Research Article
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aksoylu, B. and Holst, M., Optimality of multilevel preconditioners for local mesh refinement in three dimensions, SIAM J. Numer. Anal., 44 (2006), 10051025.Google Scholar
[2]Arnold, D. N. and Brezzi, F., Mixed and non-conforming finite element methods: implementation, post-processing and error estimates, Modél. Math. Anal. Numér., 19 (1985), 735.CrossRefGoogle Scholar
[3]Bornemann, F., Erdmann, B. and Kornhuber, R., Adaptive multilevel methods in three space dimensions, Int. J. Numer. Methods. Eng., 36 (1993), 31873203.Google Scholar
[4]Brandt, A., Multi-level adaptive solutions to boundary-value problems, Math. Comp., 31 (1977), 333390.Google Scholar
[5]Brandt, A., Mccormick, S. F. and Ruge, J. W., Algebraic Multigrid (AMG) for Automatic Multi-grid Solution with Application to Geodetic Computations, Report, Inst. Comp. Studies, Colorado State University, Ft. Collins, CO, 1982.Google Scholar
[6]Brandt, A., Algebraic multigrid theory: The symmetric case, Appl. Math. Comput., 19 (1986), 2356.Google Scholar
[7]Brezzi, F., Douglas, J. Jr. and Marini, L. D., Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), 217235.Google Scholar
[8]Chen, Z., Equivalence between and multigrid algorithms for mixed and nonconforming methods for second order elliptic problems, East-West J. Numer. Math., 4 (1994), 133.Google Scholar
[9]Chen, L., Deriving the X-Z identity from auxiliary space method, in Springer Lecture Notes Series 78, Springer-Verlag, Berlin Heidelberg, 2011.Google Scholar
[10]Chen, L., Nochetto, R. H. and Xu, J., Optimal multilevel methods for graded bisection grids, Numer. Math., 120 (2011), 134.Google Scholar
[11]Chen, L., Wang, J., Wang, Y. and Ye, X., An auxiliary space multigrid preconditioner for the weak Galerkin method, Comput. Math. Appl., 70 (2015), 330344.CrossRefGoogle Scholar
[12]Cho, D., Xu, J. and Zikatanov, L., New estimates for the rate of convergence of the method of subspace corrections, Numer. Math. Theor. Meth. Appl., 1 (2008), 4456.Google Scholar
[13]Cockburn, B., Dong, B. and Guzmán, J., A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comp., 77 (2008), 18871916.Google Scholar
[14]Cockburn, B., Dubois, O., Gopalakrishnan, J. and Tan, S., Multigrid for an HDG method, IMA J. Numer. Anal., 34 (2014), 13861425.Google Scholar
[15]Cockburn, B. and Gopalakrishnan, J., A characterization of hybridized mixed methods for second order elliptic problems, SIAM J. Numer. Anal., 42 (2004), 283301.Google Scholar
[16]Cockburn, B. and Gopalakrishnan, J., New hybridization techniques, Gamm-Mitt, 2 (2005), 154183.CrossRefGoogle Scholar
[17]Cockburn, B., Gopalakrishnan, J. and Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and conforming Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47 (2009), 13191365.Google Scholar
[18]Cockburn, B., Gopalakrishnan, J. and Sayas, F. J., A projection-based error analysis of HDG methods, Math. Comp., 79 (2010), 13511367.Google Scholar
[19]Cockburn, B., Gopalakrishnan, J. and Wang, H., Locally conservative fluxes for the continuous Galerkin method, SIAM J. Numer. Anal., 45 (2007), 17421776.Google Scholar
[20]Dahmen, W. and Kunoth, A., Multilevel preconditioning, Numer. Math., 63 (1992), 315344.Google Scholar
[21]Gopalakrishnan, J., A Schwarz preconditioner for a hybridized mixed method, Comput. Methods Appl. Math., 3 (2003), 116134.Google Scholar
[22]Gopalakrishnan, J. and Tan, S., A convergent multigrid cycle for the hybridized mixed method, Numer. Linear Algebra Appl., 16 (2009), 689714.Google Scholar
[23]Li, B. and Xie, X., A two-level algorithm for the weak Galerkin discretization of diffusion problems, J. Comput. Appl. Math., 287 (2015), 179195.Google Scholar
[24]Li, B. and Xie, X., Analysis of a family of HDG methods for second order elliptic problems,J. Comput. Appl. Math., accepted; arXiv preprint arXiv:1408.5545,2014.Google Scholar
[25]Li, B. and Xie, X., BPX preconditioner for nonstandard finite element methods for diffusion problems, SIAM Numer. Anal., accepted; arXiv preprint parXiv:1410.5332v1.Google Scholar
[26]Livne, O. E., Coarsening by compatible relaxation, Numer. Linear Algebra Appl., 11 (2004), 205227.Google Scholar
[27]Mccormick, S., Fast adaptive composite gird (FAC) methods: theory for the variational case, in Springer Computing Supplementum Series 5, Springer-Verlag, Vienna, 1984.Google Scholar
[28]Mccormick, S. F. and Thomas, J. W., The fast adaptive composite grid (FAC) method for elliptic equations, Math. Comp., 46 (1986), 439456.Google Scholar
[29]Mu, L., Wang, J., Wang, Y. and Ye, X., A computational study of the weak Galerkin method for second-order elliptic equations, Numerical Algorithms, 63 (2013), 753777.CrossRefGoogle Scholar
[30]Mu, L., Wang, J. and Ye, X., A weak Galerkin finite element method with polynomial reduction, J. Comp. Appl. Math., 285 (2015), 4558.CrossRefGoogle Scholar
[31]Mu, L., Wang, J. and Ye, X., Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12 (2015), 3153.Google Scholar
[32]Raviart, P. A. and Thomas, J. M., A mixed finite element method for second order elliptic problems, in Springer Lecture Notes series 606, Springer-Verlag, New York, 1977.Google Scholar
[33]Vaněk, P., Mandel, J. and Brezina, M., Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing, 56 (1996), 179196.Google Scholar
[34]Wang, J. and Ye, X., A weak Galerkin finite element method for second-order elliptic problems, J. Comp. Appl. Math., 241 (2013), 103115.Google Scholar
[35]Wu, H. and Chen, Z., Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems, Science in China Series A: Mathematics, 49 (2006), 14051429.Google Scholar
[36]Xu, J. and Zikatanov, L., The method of alternating projections and the method of subspace corrections in Hilbert space, J. Am. Math. Soc., 15 (2002), 573597.CrossRefGoogle Scholar