A Unified Analysis of Balancing Domain Decomposition by Constraints for Discontinuous Galerkin Discretizations
Pages 1695 - 1712
Abstract
The BDDC algorithm is extended to a large class of discontinuous Galerkin (DG) discretizations of second order elliptic problems. An estimate of $C(1+\log(H/h))^2$ is obtained for the condition number of the preconditioned system where $C$ is a constant independent of $h$ or $H$ or large jumps in the coefficient of the problem. Numerical simulations are presented which confirm the theoretical results. A key component for the development and analysis of the BDDC algorithm is a novel perspective presenting the DG discretization as the sum of elementwise „local” bilinear forms. The elementwise perspective allows for a simple unified analysis of a variety of DG methods and leads naturally to the appropriate choice for the subdomainwise local bilinear forms. Additionally, this new perspective enables a connection to be drawn between the DG discretization and a related continuous finite element discretization to simplify the analysis of the BDDC algorithm.
References
[1]
P. F. Antonietti and B. Ayuso, Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: Non-overlapping case, M2AN Math. Model. Numer. Anal., 41 (2007), pp. 21--54.
[2]
P. F. Antonietti and B. Ayuso, Class of preconditioners for discontinuous Galerkin approximation of elliptic problems, in Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng. 60, Springer, Berlin, 2008, pp. 185--192.
[3]
P. F. Antonietti and B. Ayuso, Two-level Schwarz preconditioners for super penalty discontinuous Galerkin methods, Commun. Comput. Phys., 5 (2009), pp. 398--412.
[4]
D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 1749--1779.
[5]
F. Bassi and S. Rebay, A high-order discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), pp. 267--279.
[6]
J.-F. Bourgat, R. Glowinski, P. Le Tallec, and M. Vidrascu, Variational formulation and algorithm for trace operator in domain decomposition calculations, in Domain Decomposition Methods, T. Chan, R. Glowinski, J. Periaux, O. Widlund, eds., SIAM, Philadelphia, PA, 1989, pp. 3--16.
[7]
S. C. Brenner and L.-Y. Sung, BDDC and FETI-DP without matrices or vectors, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 1429--1435.
[8]
F. Brezzi, G. Manzini, D. Marini, P. Pietra, and A. Russo, Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differential Equations, 16 (2000), pp. 365--378.
[9]
B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), pp. 2440--2463.
[10]
L. C. Cowsar, J. Mandel, and M. F. Wheeler, Balancing domain decomposition for mixed finite elements, Math. Comp., 64 (1995), pp. 989--1015.
[11]
C. R. Dohrmann, A preconditioner for substructuring based on constrained energy minimization, SIAM J. Sci. Comput., 25 (2003), pp. 246--258.
[12]
M. Dryja, J. Galvis, and M. Sarkis, BDDC methods for discontinuous Galerkin discretization of elliptic problems, J. Complexity, 23 (2007), pp. 715--739.
[13]
M. Dryja, J. Galvis, and M. Sarkis, Balancing domain decomposition methods for discontinuous Galerkin discretization, in Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng. 60, Springer, Berlin, 2008, pp. 271--278.
[14]
M. Dryja and M. Sarkis, A Neumann-Neumann method for DG discretization of elliptic problems, Tech report serie a 456, Intituto de Mathematica Pura e Aplicada, Rio de janeiro, Brazil; available online at http://www.preprint.impa.br/Shadows/SERIE_A/ 2006/456.html, 2006.
[15]
X. Feng and O. A. Karakashian, Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems, SIAM J. Numer. Anal., 39 (2001), pp. 1343--1365.
[16]
J. Li and O. B. Widlund, FETI-DP, BDDC, and block Cholesky methods, Internat. J. Numer. Methods Engrg., 66 (2006), pp. 250--271.
[17]
J. Mandel, Balancing domain decomposition, Comm. Numer. Methods Engrg., 9 (1993), pp. 233--241.
[18]
J. Mandel and C. R. Dohrmann, Convergence of a balancing domain decomposition by constraints and energy minimization, Numer. Linear Algebra Appl., 10 (2003), pp. 639--659.
[19]
J. Mandel and B. Sousedik, BDDC and FETI-DP under minimalist assumptions, Computing, 81 (2007), pp. 269--280.
[20]
J. Peraire and P.-O. Persson, The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J. Sci. Comput., 30 (2008), pp. 1806--1824.
[21]
K. Shahbazi, An explicit expression for the penalty parameter of the interior penalty method, J. Comput. Phys., 205 (2005), pp. 401--407.
[22]
A. Toselli and O. Widlund, Domain Decomposition Methods---algorithm and Theory, Springer-Verlag, Berlin, 2005.
[23]
X. Tu, A BDDC algorithm for flow in porous media with a hybrid finite element discretization, Electron. Trans. Numer. Anal., 26 (2007), pp. 146--160.
Index Terms
- A Unified Analysis of Balancing Domain Decomposition by Constraints for Discontinuous Galerkin Discretizations
Index terms have been assigned to the content through auto-classification.
Recommendations
A BDDC algorithm for the Stokes problem with weak Galerkin discretizations
AbstractThe BDDC (balancing domain decomposition by constraints) methods have been applied successfully to solve the large sparse linear algebraic systems arising from conforming finite element discretizations of second order elliptic and ...
A Numerical Study of FETI Algorithms for Mortar Finite Element Methods
The finite element tearing and interconnecting (FETI) method is an iterative substructuring method using Lagrange multipliers to enforce the continuity of the finite element solution across the subdomain interface. Mortar finite elements are nonconforming ...
BDDC algorithms for advection-diffusion problems with HDG discretizations
AbstractIn this paper, a preconditioned GMRES method is developed and analyzed for solving the linear system from advection-diffusion equations with the hybridizable discontinuous Galerkin (HDG) discretization. The preconditioner is the ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
© 2012, Society for Industrial and Applied Mathematics.
Publisher
Society for Industrial and Applied Mathematics
United States
Publication History
Published: 01 January 2012
Author Tags
Author Tags
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 07 Mar 2025