Mixed multiscale finite element method for problems in perforated media with inhomogeneous Dirichlet boundary conditions

• Vasilyeva Maria V., vasilyevadotmdotv@gmail.com M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677000, Russia
• Spiridonov Denis A., d.stalnov@mail.ru International research laboratory "Multiscale model reduction", M. K. Ammosov North-Eastern Federal University, 42 Kulakovskogo Street, Yakutsk 677000, Russia
• Chung Eric T., tschung@math.cuhk.edu.hk Department of Mathematics, The Chinese University of Hong Kong (CUHK), Hong Kong
• Efendiev Yalchin, efendiev@math.tamu.edu Department of Mathematics and Institute for Scientific Computation, Texas A&M University, College Station, TX, USA
Keywords: mixed generalized multiscale finite element method, mixed finite element method, elliptic equation, additional multiscale basis function, perforated region

Abstract

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.

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How to Cite
Vasilyeva, M., Spiridonov, D., Chung, E. and Efendiev, Y. (&nbsp;) “Mixed multiscale finite element method for problems in perforated media with inhomogeneous Dirichlet boundary conditions”, Mathematical notes of NEFU, 26(2), pp. 65-79. doi: https://doi.org/10.25587/SVFU.2019.102.31512.
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Mathematical Modeling