Skip to main content Accessibility help

Hybrid and Multiplicative Overlapping Schwarz Algorithms with Standard Coarse Spaces for Mixed Linear Elasticity and Stokes Problems

  • Mingchao Cai (a1) and Luca F. Pavarino (a2)


The goal of this work is to construct and study hybrid and multiplicative two-level overlapping Schwarz algorithms with standard coarse spaces for the almost incompressible linear elasticity and Stokes systems, discretized by mixed finite and spectral element methods with discontinuous pressures. Two different approaches are considered to solve the resulting saddle point systems: a) a preconditioned conjugate gradient (PCG) method applied to the symmetric positive definite reformulation of the almost incompressible linear elasticity system obtained by eliminating the pressure unknowns; b) a GMRES method with indefinite overlapping Schwarz preconditioner applied directly to the saddle point formulation of both the elasticity and Stokes systems. Condition number estimates and convergence properties of the proposed hybrid and multiplicative overlapping Schwarz algorithms are proven for the positive definite reformulation of almost incompressible elasticity. These results are based on our previous study [8] where only additive Schwarz preconditioners were considered for almost incompressible elasticity. Extensive numerical experiments with both finite and spectral elements show that the proposed overlapping Schwarz preconditioners are scalable, quasi-optimal in the number of unknowns across individual subdomains and robust with respect to discontinuities of the material parameters across subdomains interfaces. The results indicate that the proposed preconditioners retain a good performance also when the quasi-monotonicity assumption, required by the available theory, does not hold.


Corresponding author

*Corresponding author. Email addresses: (M. Cai), (L. F. Pavarino)


Hide All
[1] Beirão da Veiga, L., Lovadina, C. and Pavarino, L. F., Positive definite balancing Neumann-Neumann preconditioners for nearly incompressible elasticity, Numer. Math., 104 (2006), pp. 271296.
[2] Barker, A. T. and Cai, X.-C., Two-levelNewton and hybrid Schwarz preconditioners for fluid-structure interaction, SIAM J. Sci. Comp., 32 (2010), pp. 23952417.
[3] Beirão da Veiga, L., Cho, D., Pavarino, L. F., Scacchi, S., Isogeometric Schwarz preconditioners for linear elasticity systems, Comput. Meth. Appl. Mech. Engrg., 253 (2013), pp. 439454.
[4] Bernardi, C. and Maday, Y.. Spectral Methods. In Ciarlet, P. G., Lions, J. L., Eds., Handbook of Numerical Analysis Vol. 5, pp. 209485, 2007.
[5] Bernardi, C. and Maday, Y.. Uniform inf–sup conditions for the spectral discretization of the Stokes problem. Math. Models Methods Appl. Sci., 9(03):395414, 1999.
[6] Boffi, D., Gastaldi, L., On the quadrilateral Q 2P 1 element for the Stokes problem, Int. J. Numer. Meth. Fluids., 39 (11) (2002), pp. 10011011.
[7] Boffi, D., Brezzi, F. and Fortin, M., Mixed Finite Element Methods and Applications. vol. 44 of Springer Series in Computational Mathematics, Springer-Verlag, 2013.
[8] Cai, M., Pavarino, L., Widlund, O., Overlapping Schwarz methods with a standard coarse space for almost incompressible linear elasticity, SIAM, J. Sci. Comput., 37 (2) (2015), pp. A811–A831.
[9] Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A.. Spectral Methods. Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, 2007.
[10] Dobrowolski, M., On the LBB constant on stretched domains, Math. Nachr., 254 (1) (2003), pp. 6467.
[11] Dohrmann, C. R., Widlund, O. B., An overlapping Schwarz algorithm for almost incompressible elasticity, SIAM J. Numer. Anal., 47 (4) (2009), pp. 28972923.
[12] Dohrmann, C. R., Widlund, O. B., Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity, Int. J. Numer. Meth. Eng., 82 (2) (2010), pp. 157183.
[13] Dohrmann, C. R. and Widlund, O. B., A BDDC algorithm with deluxe scaling for three-dimensional H(curl) problems, Comm. Pure Appl. Math., 69 (4) (2016), pp. 745770.
[14] Dryja, M., Sarkis, M. V., and Widlund, O. B., Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions, Numer. Math., 72 (1996), pp. 313348.
[15] Hwang, F. N. and Cai, X.-C., A class of parallel two-level nonlinear Schwarz preconditioned inexact Newton algorithms, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 16031611.
[16] Fischer, P. F., An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations, J. Comput. Phys., 133 (1997), pp. 84101.
[17] Girault, V. and Raviart, P. A., Finite element methods for Navier-Stokes Equations, theory and algorithms, Springer-Verlag, Berlin, 1986.
[18] Griebel, M. and Oswald, P., On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math., 70(2) (1995), pp. 163180.
[19] Goldfeld, P., Pavarino, L. F., and Widlund, O. B., Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity, Numer. Math., 95 (2003), pp. 283324.
[20] Kim, H. H. and Lee, C.-O., A two-level nonoverlapping Schwarz algorithm for the Stokes problem without primal pressure unknowns, Internat. J. Numer. Methods Engrg., 88 (2011), pp. 13901410.
[21] Klawonn, A., Pavarino, L. F., Overlapping Schwarz methods for mixed linear elasticity and Stokes problems, Comput. Methods Appl. Mech. Engrg., 165 (1) (1998), pp. 233245.
[22] Klawonn, A., Rheinbach, O., and Wohlmut, B., Dual-primal iterative substructuring for almost incompressible elasticity, Domain decomposition methods in science and engineering XVI, vol. 55 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2007, pp. 397404.
[23] Klawonn, A., Pavarino, L. F., A comparison of overlapping Schwarz methods and block preconditioners for saddle point problems, Numer. Linear. Algebr., 7 (1) (2000), pp. 125.
[24] Le Tallec, P., Patra, A., Non-overlapping domain decomposition methods for adaptive hp approximations of the Stokes problem with discontinuous pressure fields, Comput. Meth. Appl. Mech. Engrg., 145 (3) (1997), pp. 361379.
[25] Li, J., A dual-primal FETI method for incompressible Stokes equations, Numer. Math., 102 (2005), pp. 257275.
[26] Li, J. and Widlund, O. B., BDDC algorithms for incompressible Stokes equations, SIAM J. Numer. Anal., 44 (2006), pp. 24322455.
[27] Maday, Y., Meiron, D., Patera, A. T., and Rønquist, E. M.. Analysis of iterative methods for the steady and unsteady stokes problem: Application to spectral element discretizations. SIAM J. Sci. Comput., 14(2):310337, 1993.
[28] Mandel, J., Hybrid domain decomposition with unstructured subdomains. Contemp. Math., 157 (1994) pp. 103103.
[29] Matthies, G., Tobiska, L., The inf-sup condition for the mapped element in arbitrary space dimensions, Computing, 69 (2) (2002), pp. 119139.
[30] Nabben, R., Comparisons between multiplicative and additive schwarz iterations in domain decomposition methods, Numer. Math., 95(1) (2003) pp. 145162.
[31] Notay, Y. and Napov, A., Further comparison of additive and multiplicative coarse grid correction, Appl. Numer. Math., 65 (2013), pp. 5362.
[32] Pavarino, L. F., Preconditioned mixed spectral element methods for elasticity and Stokes problems, SIAM J. Sci. Comput., 19 (6) (1998), pp. 19411957.
[33] Pavarino, L. F., Indefinite overlapping Schwarz methods for time-dependent Stokes problems, Comput. Methods Appl. Mech. Engrg., 187 (2000), pp. 3551.
[34] Pavarino, L. F. and Widlund, O. B., Iterative substructuring methods for spectral element discretizations of elliptic systems. II. Mixed methods for linear elasticity and Stokes flow, SIAM J. Numer. Anal., 37 (2000), pp. 375402.
[35] Pavarino, L. F. and Widlund, O. B., Balancing Neumann-Neumann methods for incompressible Stokes equations, Comm. Pure Appl. Math., 55 (2002), pp. 302335.
[36] Pavarino, L. F., Widlund, O. B., Zampini, S., BDDC preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions, SIAM J. Sci. Comput., 32 (2010), pp. 36043626.
[37] Toselli, A. and Widlund, O. B., Domain Decomposition Methods - Algorithms and Theory, vol. 34 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin Heidelberg New York, 2005.
[38] Tu, X. and Li, J., A unified dual-primal finite element tearing and interconnecting approach for incompressible Stokes equations, Internat. J. Numer. Methods Engrg. 94 (2013), pp. 128149.


MSC classification

Related content

Powered by UNSILO

Hybrid and Multiplicative Overlapping Schwarz Algorithms with Standard Coarse Spaces for Mixed Linear Elasticity and Stokes Problems

  • Mingchao Cai (a1) and Luca F. Pavarino (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.