Applications of Mathematics, Vol. 63, No. 6, pp. 603-628, 2018

Parallel solution of elasticity problems using overlapping aggregations

Roman Kohut

Received May 29, 2017.   Published online June 12, 2018.

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
Classification MSC:  65F08,74S05
DOI:  10.21136/AM.2018.0142-17

[1] J. C. Andersson: Rock Mass Response to Coupled Mechanical Thermal Loading: Äspö Pillar Stability Experiment. Doctoral Thesis. Byggvetenskap, Stockholm (2007), Available at
[2] R. Blaheta: Displacement decomposition - incomplete factorization preconditioning techniques for linear elasticity problems. Numer. Linear Algebra Appl. 1 (1994), 107-128. DOI 10.1002/nla.1680010203 | MR 1277796 | Zbl 0837.65021
[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. DOI 10.1007/11666806_1 | MR 2246812 | Zbl 1142.65337
[4] R. Blaheta, P. Byczanski, O. Jakl, J. Starý: Space decomposition preconditioners and their application in geomechanics. Math. Comput. Simul. 61 (2003), 409-420. DOI 10.1016/S0378-4754(02)00096-4 | MR 1984141 | Zbl 1013.65038
[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. Zbl 0995.74066
[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. DOI 10.1016/j.ijrmms.2015.03.012
[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. MR 0803712 | Zbl 0548.65014
[8] M. Brezina, T. Manteuffel, S. McCormick, J. Ruge, G. Sanders: Towards adaptive smoothed aggregation ($\alpha$SA) for nonsymmetric problems. SIAM J. Sci. Comput. 32 (2010), 14-39. DOI 10.1137/080727336 | MR 2599765 | Zbl 1210.65075
[9] W. Hackbusch: Multi-grid Methods and Applications. Springer Series in Computational Mathematics 4, Springer, Berlin (1985). DOI 10.1007/978-3-662-02427-0 | MR 0814495 | Zbl 0595.65106
[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. DOI 10.1137/S1064827500372274 | MR 1861258 | Zbl 1036.65109
[11] A. Kolcun: Conform decomposition of cube. SSCG'94: Spring School on Computer Graphics. Comenius University, Bratislava, 1994, pp. 185-191.
[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. DOI 10.1090/conm/157/01411 | MR 1262611 | Zbl 0796.65127
[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). MR 1410757 | Zbl 0857.65126
[14] U. Trottenberg, C. W. Oosterle, A. Schüller: Multigrid. Academic Press, New York (2001). MR 1807961 | Zbl 0976.65106
[15] P. Vaněk, J. Mandel, M. Brezina: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56 (1996), 179-196. DOI 10.1007/BF02238511 | MR 1393006 | Zbl 0851.65087
[16] O. C. Zienkiewicz: The Finite Element Method. McGraw-Hill, London (1977). Zbl 0435.73072

Affiliations:   Roman Kohut, Institute of Geonics, Academy of Sciences of the Czech Republic, Studentská 17, 708 00 Ostrava, Czech Republic, e-mail:

PDF available at: