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A Generalized Multiscale Finite Element Method for poroelasticity problems II: nonlinear coupling

Brown, Donald; Vasilyeva, Maria

Authors

Donald Brown donald.brown@nottingham.ac.uk

Maria Vasilyeva vasilyevadotmdotv@gmail.com



Abstract

In this paper, we consider the numerical solution of some nonlinear poroelasticity problems that are of Biot type and develop a general algorithm for solving nonlinear coupled systems. We discuss the difficulties associated with flow and mechanics in heterogenous media with nonlinear coupling. The central issue being how to handle the nonlinearities and the multiscale scale nature of the media. To compute an efficient numerical solution we develop and implement a Generalized Multiscale Finite Element Method (GMsFEM) that solves nonlinear problems on a coarse grid by constructing local multiscale basis functions and treating part of the nonlinearity locally as a parametric value. After linearization with a Picard Iteration, the procedure begins with construction of multiscale bases for both displacement and pressure in each coarse block by treating the staggered nonlinearity as a parametric value. Using a snapshot space and local spectral problems, we construct an offline basis of reduced dimension. From here an online, parametric dependent, space is constructed. Finally, after multiplying by a multiscale partitions of unity, the multiscale basis is constructed and the coarse grid problem then can be solved for arbitrary forcing and boundary conditions. We implement this algorithm on a geometry with a linear and nonlinear pressure dependent permeability field and compute error between the multiscale solution with the fine-scale solutions.

Journal Article Type Article
Publication Date Nov 27, 2015
Journal Journal of Computational and Applied Mathematics
Print ISSN 0377-0427
Electronic ISSN 1879-1778
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 297
APA6 Citation Brown, D., & Vasilyeva, M. (2015). A Generalized Multiscale Finite Element Method for poroelasticity problems II: nonlinear coupling. Journal of Computational and Applied Mathematics, 297, doi:10.1016/j.cam.2015.11.007
DOI https://doi.org/10.1016/j.cam.2015.11.007
Keywords Multiscale, Geomechanics, Finite Elements
Publisher URL http://www.sciencedirect.com/science/article/pii/S037704271500552X
Copyright Statement Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by-nc-nd/4.0

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by-nc-nd/4.0





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