### COMPUTATIONAL MULTISCALE METHODS FOR LINEAR HETEROGENEOUS POROELASTICITY

Robert Altmann1, Eric Chung2, Roland Maier3, Daniel Peterseim3, Sai-Mang Pun4

• Received:2018-09-07 Revised:2018-12-16 Online:2020-01-15 Published:2020-01-15
• Supported by:

The authors acknowledge support from the Germany/Hong Kong Joint Research Scheme sponsored by the German Academic Exchange Service (DAAD) under the project 57334719 and the Research Grants Council of Hong Kong with reference number GCUHK405/16. Further, the authors thank the Hausdorff Institute for Mathematics in Bonn for the kind hospitality during the trimester program on multiscale problems in 2017. Daniel Peterseim also acknowledges support by the Sino-German Science Center on the occasion of the Chinese-German Workshop on Computational and Applied Mathematics in Shanghai 2017.

Robert Altmann, Eric Chung, Roland Maier, Daniel Peterseim, Sai-Mang Pun. COMPUTATIONAL MULTISCALE METHODS FOR LINEAR HETEROGENEOUS POROELASTICITY[J]. Journal of Computational Mathematics, 2020, 38(1): 41-57.

We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling. This poroelasticity problem suffers from rapidly oscillating material parameters, which calls for a thorough numerical treatment. In this paper, we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity. Therein, local corrector problems are constructed in line with the static equations, whereas we propose to consider the full system. This allows to benefit from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure. We prove the optimal first-order convergence of this method and verify the result by numerical experiments.

CLC Number:

 [1] M.A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys., 12:2(1941), 155-164.[2] D.L. Brown, J. Gedicke, and D. Peterseim, Numerical homogenization of heterogeneous fractional Laplacians. Multiscale Model. Simul., 16:3(2018), 1305-1332.[3] D.L. Brown and D. Peterseim, A multiscale method for porous microstructures. Multiscale Model. Simul., 14(2016), 1123-1152.[4] D.L. Brown and M. Vasilyeva, A generalized multiscale finite element method for poroelasticity problems I:Linear problems. J. Comput. Appl. Math., 294(2016), 372-388.[5] D.L. Brown and M. Vasilyeva, A generalized multiscale finite element method for poroelasticity problems II:Nonlinear coupling. J. Comput. Appl. Math., 297(2016), 132-146.[6] A. Caiazzo and J. Mura, Multiscale modeling of weakly compressible elastic materials in the harmonic regime and applications to microscale structure estimation. Multiscale Model. Simul., 12:2(2014), 514-537.[7] E.T. Chung, Y. Efendiev, and W.T. Leung, Constraint energy minimizing generalized multiscale finite element method. Comput. Method. Appl. M., 339(2018), 298-319.[8] E.T. Chung, Y. Efendiev, and W.T. Leung. Fast online generalized multiscale finite element method using constraint energy minimization. J. Comput. Phys., 355(2018), 450-463.[9] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978.[10] P.G. Ciarlet, Mathematical elasticity. Vol. I. North-Holland, Amsterdam, 1988.[11] Y. Efendiev, J. Galvis, and T.Y. Hou, Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys., 251(2013), 116-135.[12] C. Engwer, P. Henning, A. Målqvist, and D. Peterseim, Efficient implementation of the localized orthogonal decomposition method. ArXiv e-prints, 1602.01658, accepted for publication in Comp. Meth. Appl. Mech. Eng., 2019.[13] A. Ern and S. Meunier, A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems. ESAIM Math. Model. Numer. Anal., 43:2(2009), 353-375.[14] D. Gallistl and D. Peterseim, Computation of quasi-local effective diffusion tensors and connections to the mathematical theory of homogenization. Multiscale Model. Simul., 15:4(2017), 1530-1552.[15] F. Hellman, Gridlod. https://github.com/fredrikhellman/gridlod,2017. GitHub repository, commit 3e9cd20970581a32789aa1e21d7ff3f7e8f0b334.[16] P. Henning and D. Peterseim, Oversampling for the multiscale finite element method. Multiscale Model. Simul., 11:4(2013), 1149-1175.[17] R. Kornhuber, D. Peterseim, and H. Yserentant, An analysis of a class of variational multiscale methods based on subspace decomposition. Math. Comp., 87(2018), 2765-2774.[18] R. Kornhuber and H. Yserentant. Numerical homogenization of elliptic multiscale problems by subspace decomposition. Multiscale Model. Simul., 14:3(2016), 1017-1036.[19] A. Målqvist and A. Persson, A generalized finite element method for linear thermoelasticity. ESAIM Math. Model. Numer. Anal., 51:4(2017), 1145-1171.[20] A. Målqvist and A. Persson, Multiscale techniques for parabolic equations. Numer. Math., 138:1(2018), 191-217.[21] A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems. Math. Comp., 83:290(2014), 2583-2603.[22] J. Mura and A. Caiazzo, A Two-Scale Homogenization Approach for the Estimation of Porosity in Elastic Media. Springer International Publishing, Cham, 2016, 89-105.[23] D. Peterseim, Variational multiscale stabilization and the exponential decay of fine-scale correctors. In Building Bridges:Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Springer, 2016, 341-367.[24] D. Peterseim and R. Scheichl, Robust numerical upscaling of elliptic multiscale problems at high contrast. Comput. Methods Appl. Math., 16:4(2016), 579-603.[25] R.E. Showalter, Diffusion in poro-elastic media. J. Math. Anal. Appl., 251:1(2000), 310-340.[26] M.D. Zoback, Reservoir Geomechanics. Cambridge University Press, Cambridge, 2010.
 [1] Zhongyi Huang, Xu Yang. TAILORED FINITE CELL METHOD FOR SOLVING HELMHOLTZ EQUATION IN LAYERED HETEROGENEOUS MEDIUM [J]. Journal of Computational Mathematics, 2012, 30(4): 381-391.
Viewed
Full text

Abstract