Applications of Mathematics

, Volume 63, Issue 6, pp 603–628

# Parallel solution of elasticity problems using overlapping aggregations

• Roman Kohut
Article

## Abstract

The finite element (FE) solution of geotechnical elasticity problems leads to the solution of a large system of linear equations. For solving the system, we use the preconditioned conjugate gradient (PCG) method with two-level additive Schwarz preconditioner. The preconditioning is realised in parallel. A coarse space is usually constructed using an aggregation technique. If the finite element spaces for coarse and fine problems on structural grids are fully compatible, relations between elements of matrices of the coarse and fine problems can be derived. By generalization of these formulae, we obtain an overlapping aggregation technique for the construction of a coarse space with smoothed basis functions. The numerical tests are presented at the end of the paper.

## Keywords

conjugate gradients aggregation Schwarz method finite element method geotechnical application elasticity

65F08 74S05

## References

1. [1]
2. [2]
R. Blaheta: Displacement decomposition—incomplete factorization preconditioning techniques for linear elasticity problems. Numer. Linear Algebra Appl. 1 (1994), 107–128.
3. [3]
R. Blaheta: Algebraic multilevel methods with aggregations: An overview. Large–Scale Scientific Computing (I. Lirkov et al., eds.). Lecture Notes in Computer Science 3743, Springer, Berlin, 2006, pp. 3–14.Google Scholar
4. [4]
R. Blaheta, P. Byczanski, O. Jakl, J. Starý: Space decomposition preconditioners and their application in geomechanics. Math. Comput. Simul. 61 (2003), 409–420.
5. [5]
R. Blaheta, O. Jakl, R. Kohut, J. Starý: Iterative displacement decomposition solvers for HPC in geomechanics. Large–Scale Scientific Computations of Engineering and Environmental Problems II (M. Griebel et al., eds.). Notes on Numerical Fluid Mechanics 73, Vieweg, Braunschweig, 2000, pp. 347–356.
6. [6]
R. Blaheta, R. Kohut, A. Kolcun, K. Souček, L. Staš, L. Vavro: Digital image based numerical micromechanics of geocomposites with application to chemical grouting. Int. J. Rock Mech. Min. Sci 77 (2015), 77–88.
7. [7]
A. Brandt, S. F. McCormick, J. W. Ruge: Algebraic multigrid (AMG) for sparse matrix equations. Sparsity and Its Applications (D. J. Evans, ed.). Cambridge University Press, Cambridge, 1985, pp. 257–284.
8. [8]
M. Brezina, T. Manteuffel, S. McCormick, J. Ruge, G. Sanders: Towards adaptive smoothed aggregation (SA) for nonsymmetric problems. SIAM J. Sci. Comput. 32 (2010), 14–39.
9. [9]
W. Hackbusch: Multi–grid Methods and Applications. Springer Series in Computational Mathematics 4, Springer, Berlin, 1985.
10. [10]
E. W. Jenkins, C. E. Kees, C. T. Kelley, C. T. Miller: An aggregation–based domain decomposition preconditioner for groundwater flow. SIAM J. Sci. Comput. 23 (2001), 430–441.
11. [11]
A. Kolcun: Conform decomposition of cube. SSCG’94: Spring School on Computer Graphics. Comenius University, Bratislava, 1994, pp. 185–191.Google Scholar
12. [12]
J. Mandel: Hybrid domain decomposition with unstructured subdomains. Domain Decomposition Methods in Science and Engineering (A. Quarteroni et al., eds.). Contemporary Mathematics 157, American Mathematical Society, Providence, 1994, pp. 103–112.Google Scholar
13. [13]
B. F. Smith, P. E. Bjørstad, W. D. Groop: Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge, 1996.Google Scholar
14. [14]
U. Trottenberg, C. W. Oosterle, A. Schüller: Multigrid. Academic Press, New York, 2001.Google Scholar
15. [15]
P. Vaněk, J. Mandel, M. Brezina: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56 (1996), 179–196.
16. [16]
O. C. Zienkiewicz: The Finite Element Method. McGraw–Hill, London, 1977.