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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.

The results of numerical and experimental studies of the effective thermal conductivity of dis� persed materials with contrast inclusions are presented. The numerical analysis was based on using the mul� tiscale finite element method making it possible to calculate heat transport in media with ordered and chaotic location of the inclusions. The influence of the concentration of inclusions and the manner of their packing on the value of the effective coefficient of thermal conductivity was analyzed. The coefficient of thermal con� ductivity was studied experimentally for two media, gypsum and plexiglass with inclusions of steel balls (d = 3 mm) with different packing. The comparison of the results of measurements and numerical simulation makes it possible to verify the elaborated program complex and determine the boundaries of application of the available analytical dependences.

Stabilized finite element methods application for the solution of of convection - diffusion problems is investigated. Features of algorithms of SUPG and RFB methods are discussed. Restrictions in application of these methods at the solution of filtration and adsorption problems are analyzed.