Mixed multiscale finite element method for problems in perforated media with inhomogeneous Dirichlet boundary conditions
We consider the solution of an elliptic equation in mixed formulation in a perforated medium with inhomogeneous Dirichlet boundary conditions at the perforation boundary. To solve the problem on a fine grid (reference solution), the mixed finite element method (Mixed FEM) is used, where the approximation of speed is implemented using Raviart-Thomas elements of the smallest order and piecewise constant basis functions for pressure. The solution on a coarse grid was obtained with the use of the mixed generalized multiscale finite element method (Mixed GMsFEM). Since the perforations have a great influence on the processes in the medium, it is necessary to calculate an additional basis, taking into account the effect of perforations on the solution. The article presents the results of a numerical experiment in a two-dimensional domain which confirm the efficiency of the proposed multiscale method.
 Le Bris C., Legoll F., and Lozinski A., “An MsFEM type approach for perforated domains,” Multiscale Model. Simul., 12, No. 3, 1046–1077 (2014).
 Jikov V. V., Kozlov S. M., and Oleinik O. A., Homogenization of Differential Operators and Integral Functionals, Springer, Berlin (1994).
 Allaire G. and Brizzi R., “A multiscale finite element method for numerical homogenization,” Multiscale Model. Simul., 4, No. 3, 790–812 (2005).
 Chung E. T., Iliev O., and Vasilyeva M. V., “Generalized multiscale finite element method for non-Newtonian fluid flow in perforated domain,” in: AIP Conf. Proc., 1773, No. 1, pp. 100001, AIP Publ. (2016).
 Vasilyeva M. and Tyrylgin A., “Machine learning for accelerating effective property prediction for poroelasticity problem in stochastic media,” arXiv preprint arXiv:1810.01586 (2018).
 Vasilyeva M. and Tyrylgin A., “Convolutional neural network for fast prediction of the effective properties of domains with random inclusions,” J. Phys., Conf. Ser., 1158, No. 4, 042034 (2019).
 Chung E. T., Efendiev Y., Li G., and Vasilyeva M., “Generalized multiscale finite element methods for problems in perforated heterogeneous domains,” Appl. Anal., 95, No. 10, 2254–2279 (2016).
 Chung E. T., Vasilyeva M., and Wang Y., “A conservative local multiscale model reduction technique for Stokes flows in heterogeneous perforated domains,” J. Comput. Appl. Math., 321, 389–405 (2017).
 Vasilyeva M. V. and Stalnov D. A., “Numerical homogenization for the heat problem in heterogeneous and perforated media [in Russian],” Vestn. Sev.-Vost. Feder. Univ., No. 2, 49–59 (2017).
 Tyrylgin A., Spiridonov D., and Vasilyeva M., “Numerical homogenization for poroelasticity problem in heterogeneous media,” J. Phys., Conf. Ser., 1158, No. 4, 042030 (2019).
 Chung E. T., Efendiev Y., and Lee C. S., “Mixed generalized multiscale finite element methods and applications,” Multiscale Model. Simul., 13, No. 1, 338–366 (2015).
 Chung E. T., Leung W. T., Vasilyeva M., and Wang Y., “Multiscale model reduction for transport and flow problems in perforated domains,” J. Comput. Appl. Math., 330, 519–535 (2018).
 Chung E. T., Leung W. T., and Vasilyeva M., “Mixed GMsFEM for second order elliptic problem in perforated domains,” J. Comput. Appl. Math., 304, 84–99 (2016).
 Spiridonov D., Vasilyeva M., and Leung W. T., “A Generalized Multiscale Finite Element Method (GMsFEM) for perforated domain flows with Robin boundary conditions,” J. Comput. Appl. Math., 357, 319–328 (2019).
 Vasilyeva M., Chung E. T., Leung W. T., Wang Y., and Spiridonov D., “Upscaling method for problems in perforated domains with non-homogeneous boundary conditions on perforations using Non-Local Multi-Continuum method (NLMC),” J. Comput. Appl. Math., 357, 215–227 (2019).
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