Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 6, pp. 2081-2108.

Two non-overlapping domain decomposition methods are presented for the mixed finite element formulation of linear elasticity with weakly enforced stress symmetry. The methods utilize either displacement or normal stress Lagrange multiplier to impose interface continuity of normal stress or displacement, respectively. By eliminating the interior subdomain variables, the global problem is reduced to an interface problem, which is then solved by an iterative procedure. The condition number of the resulting algebraic interface problem is analyzed for both methods. A multiscale mortar mixed finite element method for the problem of interest on non-matching multiblock grids is also studied. It uses a coarse scale mortar finite element space on the non-matching interfaces to approximate the trace of the displacement and impose weakly the continuity of normal stress. A priori error analysis is performed. It is shown that, with appropriate choice of the mortar space, optimal convergence on the fine scale is obtained for the stress, displacement, and rotation, as well as some superconvergence for the displacement. Computational results are presented in confirmation of the theory of all proposed methods.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2019057
Classification : 65N30, 65N55, 65N12, 74G15
Mots-clés : Domain decomposition, mixed finite elements, mortar finite elements, multiscale methods, linear elasticity
Khattatov, Eldar 1 ; Yotov, Ivan 1

1 
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     title = {Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry},
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Khattatov, Eldar; Yotov, Ivan. Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 6, pp. 2081-2108. doi : 10.1051/m2an/2019057. http://www.numdam.org/articles/10.1051/m2an/2019057/

[1] M. Amara and J.M. Thomas, Equilibrium finite elements for the linear elastic problem. Numer. Math. 33 (1979) 367–383. | DOI | MR | Zbl

[2] I. Ambartsumyan, E. Khattatov, J.M. Nordbotten and I. Yotov, A multipoint stress mixed finite element method for elasticity on quadrilateral grids. Preprint arXiv:1811.01928 [math.NA] (2018). | MR

[3] I. Ambartsumyan, E. Khattatov, J.M. Nordbotten and I. Yotov, A multipoint stress mixed finite element method for elasticity on simplicial grids. Preprint arXiv:1805.09920 [math.NA] (2018). | MR

[4] T. Arbogast, L.C. Cowsar, M.F. Wheeler and I. Yotov, Mixed finite element methods on nonmatching multiblock grids. SIAM J. Numer. Anal. 37 (2000) 1295–1315. | DOI | MR | Zbl

[5] T. Arbogast, G. Pencheva, M.F. Wheeler and I. Yotov, A multiscale mortar mixed finite element method. Multiscale Model. Simul. 6 (2007) 319–346. | DOI | MR | Zbl

[6] D. Arndt, W. Bangerth, D. Davydov, T. Heister, L. Heltai, M. Kronbichler, M. Maier, J.-P. Pelteret, B. Turcksin and D. Wells, The deal.II library, version 8.5. J. Numer. Math. 25 (2017) 137–146. | DOI | MR | Zbl

[7] D.N. Arnold and J.J. Lee, Mixed methods for elastodynamics with weak symmetry. SIAM J. Numer. Anal. 52 (2014) 2743–2769. | DOI | MR | Zbl

[8] D.N. Arnold, F. Brezzi and J. Douglas Jr, PEERS: a new mixed finite element for plane elasticity. Japan J. Appl. Math. 1 (1984) 347–367. | DOI | MR | Zbl

[9] D.N. Arnold, R.S. Falk and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput. 76 (2007) 1699–1723. | DOI | MR | Zbl

[10] D.N. Arnold, G. Awanou and W. Qiu, Mixed finite elements for elasticity on quadrilateral meshes. Adv. Comput. Math. 41 (2015) 553–572. | DOI | MR | Zbl

[11] G. Awanou, Rectangular mixed elements for elasticity with weakly imposed symmetry condition. Adv. Comput. Math. 38 (2013) 351–367. | DOI | MR | Zbl

[12] D. Boffi, F. Brezzi and M. Fortin, Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8 (2009) 95–121. | DOI | MR | Zbl

[13] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. In: Vol. 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York (1991). | DOI | MR | Zbl

[14] P.G. Ciarlet, The finite element method for elliptic problems. In: Vol. 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA (2002). | MR | Zbl

[15] B. Cockburn, J. Gopalakrishnan and J. Guzmán, A new elasticity element made for enforcing weak stress symmetry. Math. Comput. 79 (2010) 1331–1349. | DOI | MR | Zbl

[16] L.C. Cowsar, J. Mandel and M.F. Wheeler, Balancing domain decomposition for mixed finite elements. Math. Comput. 64 (1995) 989–1015. | DOI | MR | Zbl

[17] M. Dauge, Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions. In: Vol. 1341 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1988). | MR | Zbl

[18] Y. Efendiev, J. Galvis and T.Y. Hou, Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys. 251 (2013) 116–135. | DOI | MR | Zbl

[19] C. Farhat and F.-X. Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Methods Eng. 32 (1991) 1205–1227. | DOI | MR | Zbl

[20] A. Fritz, S. Hüeber and B.I. Wohlmuth, A comparison of mortar and Nitsche techniques for linear elasticity. Calcolo 41 (2004) 115–137. | DOI | MR | Zbl

[21] J. Galvis and M. Sarkis, Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. Electron. Trans. Numer. Anal. 26 (2007) 350–384. | MR | Zbl

[22] B. Ganis and I. Yotov, Implementation of a mortar mixed finite element method using a multiscale flux basis. Comput. Methods Appl. Mech. Eng. 198 (2009) 3989–3998. | DOI | MR | Zbl

[23] V. Girault, G.V. Pencheva, M.F. Wheeler and T.M. Wildey, Domain decomposition for linear elasticity with DG jumps and mortars. Comput. Methods Appl. Mech. Eng. 198 (2009) 1751–1765. | DOI | MR | Zbl

[24] R. Glowinski and M.F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, edited by R. Glowinskin, G.H. Golub, G.A. Meurant and J. Périaux. In: First International Symposium on Domain Decomposition Methods for Partial Differential Equations. SIAM, Philadelphia (1988) 144–172. | MR | Zbl

[25] P. Goldfeld, L.F. Pavarino and O.B. Widlund, Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity. Numer. Math. 95 (2003) 283–324. | DOI | MR | Zbl

[26] J. Gopalakrishnan and J. Guzmán, A second elasticity element using the matrix bubble. IMA J. Numer. Anal. 32 (2012) 352–372. | DOI | MR | Zbl

[27] P. Grisvard, Elliptic problems in nonsmooth domains. In: Vol. 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2011). | MR | Zbl

[28] P. Hauret and P. Le Tallec, A discontinuous stabilized mortar method for general 3D elastic problems. Comput. Methods Appl. Mech. Eng. 196 (2007) 4881–4900. | DOI | MR | Zbl

[29] C.T. Kelley, Iterative methods for linear and nonlinear equations. In: Vol. 16 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1995). | MR | Zbl

[30] H.H. Kim, A BDDC algorithm for mortar discretization of elasticity problems. SIAM J. Numer. Anal. 46 (2008) 2090–2111. | DOI | MR | Zbl

[31] H.H. Kim, A FETI-DP formulation of three dimensional elasticity problems with mortar discretization. SIAM J. Numer. Anal. 46 (2008) 2346–2370. | DOI | MR | Zbl

[32] A. Klawonn and O.B. Widlund, A domain decomposition method with Lagrange multipliers for linear elasticity. In: Eleventh International Conference on Domain Decomposition Methods (London, 1998). Augsburg (1999) 49–56. | MR

[33] J. Kovacik, Correlation between Young’s modulus and porosity in porous materials. J. Mater. Sci. Lett. 18 (1999) 1007–1010. | DOI

[34] J.-L. Lions and E. Magenes, In: Vol. I of Non-homogeneous Boundary Value Problems and Applications. Springer-Verlag, New York-Heidelberg (1972). | MR | Zbl

[35] T.P. Mathew, Domain decomposition and iterative refinement methods for mixed finite element discretizations of elliptic problems. Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University (1989), Tech. Rep. 463. | MR

[36] L.F. Pavarino, O.B. Widlund and S. Zampini, BDDC preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions. SIAM J. Sci. Comput. 32 (2010) 3604–3626. | DOI | MR | Zbl

[37] G. Pencheva and I. Yotov, Balancing domain decomposition for mortar mixed finite element methods. Numer. Linear Algebra Appl. 10 (2003) 159–180. | DOI | MR | Zbl

[38] G.V. Pencheva, M. Vohralk, M.F. Wheeler and T. Wildey, Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling. SIAM J. Numer. Anal. 51 (2013) 526–554. | DOI | MR | Zbl

[39] M. Peszyńska, M.F. Wheeler and I. Yotov, Mortar upscaling for multiphase flow in porous media. Comput. Geosci. 6 (2002) 73–100. | DOI | MR | Zbl

[40] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Clarendon Press, Oxford (1999). | MR | Zbl

[41] J.E. Roberts and J.-M. Thomas, Mixed and hybrid methods. In: Vol. II of Handbook of Numerical Analysis. North-Holland, Amsterdam (1991) 523–639. | DOI | MR | Zbl

[42] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. | DOI | MR | Zbl

[43] R. Stenberg, A family of mixed finite elements for the elasticity problem. Numer. Math. 53 (1988) 513–538. | DOI | MR | Zbl

[44] A. Toselli and O. Widlund, Domain decomposition methods – algorithms and theory. In: Vol. 34 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (2005). | DOI | MR | Zbl

[45] D. Vassilev, C. Wang and I. Yotov, Domain decomposition for coupled Stokes and Darcy flows. Comput. Methods Appl. Mech. Eng. 268 (2014) 264–283. | DOI | MR | Zbl

[46] M.F. Wheeler and I. Yotov, A posteriori error estimates for the mortar mixed finite element method. SIAM J. Numer. Anal. 43 (2005) 1021–1042. | DOI | MR | Zbl

[47] M.F. Wheeler, G. Xue and I. Yotov, A multiscale mortar multipoint flux mixed finite element method. ESAIM Math. Model. Numer. Anal. 46 (2012) 759–796. | DOI | Numdam | MR | Zbl

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