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.