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We present mathematical simulating results of a gas-liquid mixture flow in a cylindrical pipe. The gas-liquid mixture consists of the mineralized water and propane gas. To analyze the mineral composition of the fluid, a sensor is mounted in the pipe. The sensor affects the topology of the flow velocity vector field. The gas-liquid mixture flow is described by the unsteady Navier-Stokes equations. When the flow rate of gas-liquid mixtures exceeds 20 m/s, eddy flows occur in the pipe with obstacles. A method of discretization should take into account the problem specifics: rapidly changing gradients, the prevalence of the convective term in the Navier-Stokes equations. A computational scheme of discontinuous Galerkin method has the local conservative property and is best suited for solving such singularly perturbed problems. To perform a spatial discretization, a computational scheme of the discontinuous Galerkin method in the function spaces H(div) and L 2 is used. Application of the multiscale approach allows breaking down the solution of the simulation problem into several smaller ones that can be solved using parallel computations. Mathematical modeling results of the gas-liquid mixture flow in the pipe with different options for the sensor location are presented.

In the edge vector finite element solution of the frequency domain Maxwell equations, the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers. We extend the approach of [R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1) (1999) 204–225] and, using domain decomposition ideas, construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements.The new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain Maxwell equations. The solver can be seen as a variable preconditioner and, thus, accelerated by Krylov subspace techniques (e.g. GCR or FGMRES). We demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity.

A problem of electrical potential distribution in media, containing micro-inclusions with highly contrasting physical properties and different geometric shapes, was considered. Multiscale finite element method has been chosen as a solver. A procedure of homogenization of electrical resistivity of heterogeneous media was held, using the solution of original task. The results of numerical modeling were compared with the results of physical experiment.