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In this paper, we propose a numerical method to obtain an effective electrical resistivity of heterogeneous media under the influence of a direct current. The heterogeneous multiscale finite element method is used to solve the direct problem of simulation of an electrostatic field. The computational experiments using the developed software complex showed that even the small inclusion concentrations define the effective resistivity of the media. In addition, the change in the localization, orientation, and geometrical shape of inclusions also leads to a significant change of the effective properties of the media.

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.

A method for three-dimensional modeling of pulsed soundings using a fast Fourier transform is developed: instead of the time-domain problem, it is proposed to solve a set of frequency-domain problems obtained by processing the original excitation pulse by a fast Fourier transform. The error of the proposed method compared to the method of solving the time-domain problem is numerically estimated for a model problem with a sinusoidal signal.

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 new implementation of the Godunov-inverse iteration, that is, inverse iteration with guaranteed accuracy, is presented. A new implementation of the Cullum-Willoughby-Lanczos method, that is the Lanczos method, with guaranteed accuracy for computing the partial spectral decomposition of real symmetric matrices and the partial singular value decomposition of large sparse nonsymmetric matrices, is also presented. An inverse iteration with guaranteed accuracy is used for estimating the accuracy and, if necessary, for computing eigenvectors and singular vectors.

A mixed variational problem formulation is proposed to find the electric field as a solution to a second-order differential equation and the magnetic flux density vector as a solution to a first-order differential equation.

A new variational statement represented in the form of computational algorithm is proposed for elliptic systems of equations, which describe the behavior of structurally inhomogeneous geomedia under static loads. The inverse problem of boundary condition determination and calculated domain orientation from the point data of strain measurement is solved, which makes it possible to simulate adequately the stress-strain state of a rock mass.

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.

The electromagnetic methods are widely used for solving problems of surveying and defectoscopy in geophysics. The method of high-frequency induction logging isoparametric sounding (VIKIZ) [1] is directed toward reconstructing resistivity spatial distribution of rock, where oil- and gas-bearing wells are situated.