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We consider a multispecies competition model in a one- and two-dimensional formulation. To solve the problem numerically, we construct a discrete system using finite volume approximation by space with semi-implicit time approximation. The solution of the multispecies competition model converges to the final equilibrium state that does not depend on the initial condition of the system. The final equilibrium state characterizes the survival status of the multispecies system (one or more species survive or no one survives). In real-world problems values of the parameters are unknown and vary in some range. For such problems, the series of Monte Carlo simulations can be used to estimate the system, where a large number of simulations are needed to be performed with random values of the parameters. A numerical solution is expensive, especially for high-dimensional problems, and requires a large amount of time to perform. In this work, to reduce the cost of simulations, we use a deep neural network to fast predict the survival status. Numerical results are presented for different neural network configurations. The comparison with convenient classifiers is presented. 

In this paper, we consider a coupled system of equations that describes simplified magnetohydrodynamics (MHD) problem in perforated domains. We construct a fine grid that resolves the perforations on the grid level in order to use a traditional approximation. For the solution on the fine grid, we construct approximation using the mixed finite element method. To reduce the size of the fine grid system, we will develop a Mixed Generalized Multiscale Finite Element Method (Mixed GMsFEM). The method differs from existing approaches and requires some modifications to represent the flow and magnetic fields. Numerical results are presented for a two-dimensional model problem in perforated domains. This model problem is a special case for the general 3D problem. We study the influence of the number of multiscale basis functions on the accuracy of the method and show that the proposed method provides a good accuracy with few basis functions. 

In this paper, we consider unsaturated filtration in heterogeneous porous media with rough surface topography. The surface topography plays an important role in determining the flow process and includes multiscale features. The mathematical model is based on the Richards’ equation with three different types of boundary conditions on the surface: Dirichlet, Neumann, and Robin boundary conditions. For coarse-grid discretization, the Generalized Multiscale Finite Element Method (GMsFEM) is used. Multiscale basis functions that incorporate small scale heterogeneities into the basis functions are constructed. To treat rough boundaries, we construct additional basis functions to take into account the influence of boundary conditions on rough surfaces. We present numerical results for two-dimensional and three-dimensional model problems. To verify the obtained results, we calculate relative errors between the multiscale and reference (fine-grid) solutions for different numbers of multiscale basis functions. We obtain a good agreement between fine-grid and coarse-grid solutions.