A multigrid finite element method for 2-D convection-heat transfer problem is proposed. The method is based on minimization of the energy functional. Effectiveness of the method is demonstrated by calculation of temperature distribution in a microchip that separates living cells from fluid stream by electrophoresis.Keywords: mathematical modeling, energy method, elliptical equation, nonsymmetric operator, electrophoresisThe problem of heat transfer in a moving medium and similar convection-diffusion problem arise in various fields of research, for example, in mathematical modeling of the transport of pollutants and energy in the atmosphere or in mathematical modeling of devices created for biological research.Heat transfer is described by second order elliptic equation with proportional to velocity first order term. Then operators of boundary value problems are asymmetric operators. Because of this, the principle of minimum of a quadratic energy functional is not formulated. For problems with symmetric operators this principle allows one to use the most effective methods for the approximate and numerical solutions [10]. For problems with asymmetric operator the principles of nonequilibrium thermodynamics of minimal entropy production are not formulated, etc. [11]. Some problems with asymmetric operators are reformulated as problems with symmetric positive definite operators, and for them the energy principles are proved in [2,4]. Three-dimensional problems are discussed in [3].The aim of this work is to create a multigrid finite element method for solving two-dimensional heat transfer problem in a moving medium and to demonstrate effectiveness of the method.As an example temperature distribution in the microchip, that separates living cells from a fluid stream by means of electrophoresis is calculated.

Standard models of ionospheric delays have errors of order 1-8 TECU (standard total electron content units). On the basis of the free interpolation framework we propose a new simple model of the slant TEC distributions approximating slant TEC distributions obtained from the three-dimensional ionospheric models NeQuick2 and IRI-2016 with RMS error < 0.05 TECU. The proposed model was tested for varios positions of receivers in mid-latitude and equatorial regions. Stability of the coefficients of the model with respect to the position of the receiver and time is substantiated.