Schwarz Methods: To Symmetrize or Not to Symmetrize
Abstract
A preconditioning theory is presented which establishes sufficient conditions for multiplicative and additive Schwarz algorithms to yield self-adjoint positive definite preconditioners. It allows for the analysis and use of nonvariational and nonconvergent linear methods as preconditioners for conjugate gradient methods, and it is applied to domain decomposition and multigrid. It is illustrated why symmetrizing may be a bad idea for linear methods. It is conjectured that enforcing minimal symmetry achieves the best results when combined with conjugate gradient acceleration. Also, it is shown that the absence of symmetry in the linear preconditioner is advantageous when the linear method is accelerated by using the Bi-CGstab method. Numerical examples are presented for two test problems which illustrate the theory and conjectures.
MSC codes
Keywords
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
S. Ashby, M. Holst, T. Manteuffel, and P. Saylor. The role of the inner product in stopping criteria for conjugate gradient iterations. Technical Report UCRL‐JC‐112586, Lawrence Livermore National Laboratory, 1992.
2.
Steven Ashby, Thomas Manteuffel, Paul Saylor, A taxonomy for conjugate gradient methods, SIAM J. Numer. Anal., 27 (1990), 1542–1568
3.
Randolph Bank, Mohamed Benbourenane, The hierarchical basis multigrid method for convection‐diffusion equations, Numer. Math., 61 (1992), 7–37
4.
James Bramble, Joseph Pasciak, Jun Wang, Jinchao Xu, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comp., 57 (1991), 23–45
5.
James Bramble, Joseph Pasciak, Jun Wang, Jinchao Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp., 57 (1991), 1–21
6.
A. Brandt. Multi‐level adaptive solutions to boundary‐value problems. Math. Comp., 31, 333–390, 1977.
7.
J. M. Briggs and J. A. McCammon. Computation unravels mysteries of molecular biophysics. Computers in Physics, 6(3):238–243, 1990.
8.
P. Concus, G. H. Golub, and D. P. O’Leary. A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. In J. R. Bunch and D. J. Rose, editors, Sparse Matrix Computations, pages 309–332. Academic Press, New York, NY, 1976.
9.
, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, Proceedings of the symposium held in Houston, Texas, March 20–22, 1989, Society for Industrial and Applied Mathematics (SIAM), 1990, 0–0, xx+491
10.
R. P. Fedorenko. A relaxation method for solving elliptic difference equations. USSR Comput. Math. and Math. Phys., 1(5):1092–1096, 1961.
11.
W. Hackbusch. Multi‐grid Methods and Applications. Springer‐Verlag, Berlin, Germany, 1985.
12.
Wolfgang Hackbusch, Iterative solution of large sparse systems of equations, Applied Mathematical Sciences, Vol. 95, Springer‐Verlag, 1994xxii+429, Translated and revised from the 1991 German original
13.
Magnus Hestenes, Eduard Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1952), 409–436
14.
M. Holst. An Algebraic Schwarz Theory. Technical Report CRPC‐94‐12, Applied Mathematics and CRPC, California Institute of Technology, 1994.
15.
M. Holst and F. Saied. Multigrid and domain decomposition methods for electrostatics problems. In D. E. Keyes and J. Xu, editors, Domain Decomposition Methods in Science and Engineering (Proceedings of the Seventh International Conference on Domain Decomposition, October 27–30, 1993, The Pennsylvania State University). American Mathematical Society, Providence, 1995.
16.
, Domain decomposition methods in science and engineering, Proceedings of the Sixth International Conference on Domain Decomposition held in Como, June 15–19, 1992, Contemporary Mathematics, Vol. 157, American Mathematical Society, 1994, 0–0, xxii+484
17.
R. Kress. Linear Integral Equations. Springer‐Verlag, Berlin, Germany, 1989.
18.
Erwin Kreyszig, Introductory functional analysis with applications, Wiley Classics Library, John Wiley & Sons Inc., 1989xvi+688
19.
S. McCormick, J. Ruge, Unigrid for multigrid simulation, Math. Comp., 41 (1983), 43–62
20.
J. M. Ortega. Numerical Analysis: A Second Course. Academic Press, New York, NY, 1972.
21.
Youcef Saad, Martin Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), 856–869
22.
H. A. Schwarz. Über einige Abbildungsaufgaben. Ges. Math. Abh., 11:65–83, 1869.
23.
Peter Sonneveld, CGS, a fast Lanczos‐type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 10 (1989), 36–52
24.
E. F. Van de Velde. Domain decomposition vs. concurrent multigrid. Technical Report CRPC‐94‐11, Applied Mathematics and CRPC, California Institute of Technology, 1994.
25.
H. van der Vorst, Bi‐CGSTAB: a fast and smoothly converging variant of Bi‐CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), 631–644
26.
Folkmar Bornemann, Harry Yserentant, A basic norm equivalence for the theory of multilevel methods, Numer. Math., 64 (1993), 455–476
27.
Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992), 581–613
28.
Harry Yserentant, Two preconditioners based on the multi‐level splitting of finite element spaces, Numer. Math., 58 (1990), 163–184
Information & Authors
Information
Published In
SIAM Journal on Numerical Analysis
Pages: 699 - 722
ISSN (online): 1095-7170
Copyright
Copyright © 1997 Society for Industrial and Applied Mathematics.
History
Published online: 25 July 2006
MSC codes
Keywords
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Auxiliary Space Preconditioners for a $$C^{0}$$ Finite Element Approximation of Hamilton–Jacobi–Bellman Equations with Cordes CoefficientsJournal of Scientific Computing, Vol. 92, No. 3 | 3 August 2022
- Schwarz methods by domain truncationActa Numerica, Vol. 31 | 9 June 2022
- An H-Multigrid Method for Hybrid High-Order DiscretizationsSIAM Journal on Scientific Computing, Vol. 43, No. 5 | 16 September 2021
- A Combined Preconditioning Strategy for Nonsymmetric SystemsSIAM Journal on Scientific Computing, Vol. 36, No. 6 | 4 November 2014
- Schwarz preconditioned CG algorithm for the mortar finite elementNumerical Algorithms, Vol. 58, No. 2 | 10 March 2011
- Multilevel Schwarz methods for elliptic partial differential equationsComputer Methods in Applied Mechanics and Engineering, Vol. 200, No. 25-28 | 1 Jun 2011
- Approximation Theory and MethodsGreen's Functions and Boundary Value Problems | 7 March 2011
- Finite Element Analysis of the Time-Dependent Smoluchowski Equation for Acetylcholinesterase Reaction Rate CalculationsBiophysical Journal, Vol. 92, No. 10 | 1 May 2007
- Interior boundary conditions in the Schwarz alternating method for the Trefftz methodEngineering Analysis with Boundary Elements, Vol. 29, No. 5 | 1 May 2005
- Mathematics and Molecular NeurobiologyComputational Methods for Macromolecules: Challenges and Applications | 1 Jan 2002
- Adaptive multilevel finite element solution of the Poisson-Boltzmann equation I. Algorithms and examplesJournal of Computational Chemistry, Vol. 21, No. 15 | 1 January 2000
- Fourier Analysis of GMRES(m) Preconditioned by MultigridSIAM Journal on Scientific Computing, Vol. 22, No. 2 | 25 July 2006
- Iterative Linear Solvers in a 2D Radiation–Hydrodynamics Code: Methods and PerformanceJournal of Computational Physics, Vol. 154, No. 1 | 1 Sep 1999