Iterative solvers for BEM algebraic systems of equations. (English) Zbl 0946.65122
In the first part of the paper, the authors give an overview over standard Krylov subspace iterative methods such as the conjugate gradient (CG) method for symmetric and positive definite systems and its generalization to unsymmetric systems of algebraic equations. The question of preconditioning is briefly discussed for the case of unsymmetric systems.
The application of CG-like iterative methods to the solution of unsymmetric systems with dense and sometimes ill-conditioned system matrices usually arising from the boundary element collocation discretization of some boundary integral formulation of elliptic second-order boundary value problems is considered in the second part of the paper. The authors compare the convergence behaviour of several well-known CG-like methods such as CGS, Bi-CG, and Bi-CGStab with and without (simple) preconditioners. They consider 2D as well as 3D test problems.
The application of CG-like iterative methods to the solution of unsymmetric systems with dense and sometimes ill-conditioned system matrices usually arising from the boundary element collocation discretization of some boundary integral formulation of elliptic second-order boundary value problems is considered in the second part of the paper. The authors compare the convergence behaviour of several well-known CG-like methods such as CGS, Bi-CG, and Bi-CGStab with and without (simple) preconditioners. They consider 2D as well as 3D test problems.
Reviewer: U.Langer (Linz)
MSC:
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 65F10 | Iterative numerical methods for linear systems |
Keywords:
boundary integral method; conjugate gradient method; Krylov subspace iterative methods; preconditioning; unsymmetric systems; ill-conditioned system; boundary element collocation; elliptic second-order boundary value problems; convergenceReferences:
| [1] | Jackson, C. P.; Robinson, P., A numerical study of various algorithms related to the preconditioned Conjugate Gradient method, Int J Numer Methods Engineering, 21, 1315-1338 (1985) · Zbl 0576.65021 |
| [2] | Golub GH, Van Loan. Matrix computations. Baltimore, MD: Johns Hopkins University Press, 1990.; Golub GH, Van Loan. Matrix computations. Baltimore, MD: Johns Hopkins University Press, 1990. · Zbl 0865.65009 |
| [3] | Van Der Vorst HA. Parallel iterative solution methods for linear systems arising from discretized PDE’s. AGARD, Neuilly-Sur-Seine, France, 1995.; Van Der Vorst HA. Parallel iterative solution methods for linear systems arising from discretized PDE’s. AGARD, Neuilly-Sur-Seine, France, 1995. |
| [4] | Langtangen, H. P.; Tveito, A., A numerical comparison of Conjugate Gradient-like methods, Communications in Applied Numerical Methods, 4, 793-798 (1988) · Zbl 0659.65032 |
| [5] | Faber, V.; Manteuffel, T. A., Orthogonal error methods, SIAM J Num Analysis, 24, 1, 170-187 (1987) · Zbl 0613.65030 |
| [6] | Brebia CA, Telles JC, Wrobel LC. Boundary element techniques. Berlin: Springer-Verlag, 1984.; Brebia CA, Telles JC, Wrobel LC. Boundary element techniques. Berlin: Springer-Verlag, 1984. · Zbl 0556.73086 |
| [7] | Dax, A., The convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations, SIAM Review, 32, 4, 611-635 (1990) · Zbl 0718.65021 |
| [8] | Valente FP. Iterative methods for solving the systems of linear equations in BEM (in Portuguese), Thesis for Prof. Coord., Instituto Politécnico da Guarda, Portugal, 1992.; Valente FP. Iterative methods for solving the systems of linear equations in BEM (in Portuguese), Thesis for Prof. Coord., Instituto Politécnico da Guarda, Portugal, 1992. |
| [9] | Valente FP, Pina HLG. Iterative solvers for BEM algebraic systems of equations, boundary element technology VIII. Computational Mechanics Publications, Southampton, 1993.; Valente FP, Pina HLG. Iterative solvers for BEM algebraic systems of equations, boundary element technology VIII. Computational Mechanics Publications, Southampton, 1993. |
| [10] | Kane, J. H.; Keyes, D. E.; Prasad, K. G., Iterative solution techniques in Boundary Element analysis, Int J Numer Methods in Engineering, 31, 1511-1536 (1991) · Zbl 0825.73901 |
| [11] | Vatsya, S. R., Convergence of conjugate residual like methods to solve linear equations, SIAM J Num Analysis, 25, 4, 957-964 (1988) · Zbl 0651.65025 |
| [12] | Mansur, W. J.; Araújo, F. C.; Malaghini, J. E.B., Solution of BEM systems of equations via iterative techniques, Int J Numerical Methods in Engineering, 33, 1823-1841 (1992) · Zbl 0767.73085 |
| [13] | Eisenstat, S. C.; Elman, H. C.; Schultz, M. H., Variational iterative methods for nonsymmetric systems of linear equations, SIAM J Num Analysis, 20, 2, 345-357 (1983) · Zbl 0524.65019 |
| [14] | Fletcher R. Conjugate gradient methods for indefinite systems, lecture notes in mathematics, vol. 506. Berlin: Springer-Verlag, 1976:73-89.; Fletcher R. Conjugate gradient methods for indefinite systems, lecture notes in mathematics, vol. 506. Berlin: Springer-Verlag, 1976:73-89. · Zbl 0326.65033 |
| [15] | Sonneveld, P., CGS, A fast Lanczos-type solver for nonsymmetric linear systems, SIAM J Sci Stat Computat, 10, 1, 36-52 (1989) · Zbl 0666.65029 |
| [16] | Saad, Y.; Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J Sci Stat Computat, 7, 3, 856-864 (1986) · Zbl 0599.65018 |
| [17] | Saad, Y., The Lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems, SIAM J Num Analysis, 19, 3, 485-506 (1982) · Zbl 0483.65022 |
| [18] | Sleijpen, G. L.G.; Fokkema, D. R., BiCGStab(l) for linear equations envolving unsymmetric matrices with complex spectrum, Electronic Transactions on Numerical Analysis, 1, 11-32 (1993) · Zbl 0820.65016 |
| [19] | Van Der Vorst, H. A., Bi-CGStab: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J Sci Stat Computat, 13, 2, 631-644 (1992) · Zbl 0761.65023 |
| [20] | Neylon KJ. Numerical experiences with Bi-CGStab on advection-diffusion problem, numerical analysis report. University of Reading, 1993.; Neylon KJ. Numerical experiences with Bi-CGStab on advection-diffusion problem, numerical analysis report. University of Reading, 1993. |
| [21] | Brussino, G.; Sonnad, V., A comparison of direct and preconditioned iterative techniques for sparse, unsymmetric systems of linear equations, Int J Numer Methods in Engineering, 28, 801-815 (1989) |
| [22] | Van Der Vorst, H. A., High performance preconditioning, SIAM J Sci Stat Computat, 10, 6, 1174-1185 (1989) · Zbl 0693.65027 |
| [23] | Van Der Vorst, H. A.; Dekker, K., Conjugate gradient type methods and preconditioning, J of Comp and Applied Mathematics, 24, 73-87 (1988) · Zbl 0659.65033 |
| [24] | Langtangen, H. P., Conjugate Gradient methods and ILU preconditioning of non-symmetric matrix systems with arbitrary sparsity patterns, Int J Numerical Methods in Fluids, 9, 213-233 (1989) · Zbl 0664.65027 |
| [25] | Axelsson O. Iterative solution methods. Cambridge: Cambridge University Press, 1994.; Axelsson O. Iterative solution methods. Cambridge: Cambridge University Press, 1994. · Zbl 0795.65014 |
| [26] | Bird RB, Stewart WE, Lightfoot EN. Transport phenomena. New York: Wiley, 1960.; Bird RB, Stewart WE, Lightfoot EN. Transport phenomena. New York: Wiley, 1960. |
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