Previous |  Up |  Next


conjugate gradients; aggregation; Schwarz method; finite element method; geotechnical application; elasticity
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.
[1] Andersson, J. C.: Rock Mass Response to Coupled Mechanical Thermal Loading: "Aspö Pillar Stability Experiment. Doctoral Thesis. Byggvetenskap, Stockholm (2007), Available at\%3Anbn\%3Ase\%3Akth\%3Adiva-4287\kern0pt
[2] Blaheta, R.: 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] Blaheta, R.: Algebraic multilevel methods with aggregations: An overview. Large-Scale Scientific Computing Lecture Notes in Computer Science 3743, Springer, Berlin I. Lirkov et al. (2006), 3-14. DOI 10.1007/11666806_1 | MR 2246812 | Zbl 1142.65337
[4] Blaheta, R., Byczanski, P., Jakl, O., Starý, J.: 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] Blaheta, R., Jakl, O., Kohut, R., Starý, J.: Iterative displacement decomposition solvers for HPC in geomechanics. Large-Scale Scientific Computations of Engineering and Environmental Problems II Notes on Numerical Fluid Mechanics 73, Vieweg, Braunschweig M. Griebel et al. (2000), 347-356. Zbl 0995.74066
[6] Blaheta, R., Kohut, R., Kolcun, A., Souček, K., Staš, L., Vavro, L.: 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] Brandt, A., McCormick, S. F., Ruge, J. W.: Algebraic multigrid (AMG) for sparse matrix equations. Sparsity and Its Applications D. J. Evans Cambridge University Press, Cambridge (1985), 257-284. MR 0803712 | Zbl 0548.65014
[8] Brezina, M., Manteuffel, T., McCormick, S., Ruge, J., Sanders, G.: 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] Hackbusch, W.: 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] Jenkins, E. W., Kees, C. E., Kelley, C. T., Miller, C. T.: 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] Kolcun, A.: Conform decomposition of cube. SSCG'94: Spring School on Computer Graphics Comenius University, Bratislava (1994), 185-191.
[12] Mandel, J.: Hybrid domain decomposition with unstructured subdomains. Domain Decomposition Methods in Science and Engineering A. Quarteroni et al. Contemporary Mathematics 157, American Mathematical Society, Providence (1994), 103-112. DOI 10.1090/conm/157/01411 | MR 1262611 | Zbl 0796.65127
[13] Smith, B. F., Bjørstad, P. E., Groop, W. D.: Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996). MR 1410757 | Zbl 0857.65126
[14] Trottenberg, U., Oosterle, C. W., Schüller, A.: Multigrid. Academic Press, New York (2001). MR 1807961 | Zbl 0976.65106
[15] Vaněk, P., Mandel, J., Brezina, M.: 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] Zienkiewicz, O. C.: The Finite Element Method. McGraw-Hill, London (1977). Zbl 0435.73072
Partner of
EuDML logo