Kolesnikova N. The Methods of Barycentric Averaging in the Problems of Recovery Harmonic and Biharmonic Functions

The object is the problem of deformable body mechanics, electrostatics, heat engineering, theory of elasticity which are came to boundary problems of mathematical physics for the Laplace's, Poisson, and Sophie Germain equations; the subjects are geometrical models and mathematical methods for the thermal fields, torsion and bending strains fields built on them; the purpose is effectiveness increase of technical systems computational modelling by means of scalar and vector physical fields mathematical models for the recovery of harmonic and biharmonic multivariable functions using methods of barycentric averaging problems development; methods - methods of barycentric averaging, interpolation technique, ap-proximation approach, finite difference method, finite element analysis, probability theory, method of geometrical modeling, mathematical physics methods, least-squares method; novelty - triangular finite Hermite's type element which models elastic plate bending strains, what gives an opportunity to exclude linear algebraic equations systems generation and solution bulky procedures from the computational algorithm basis functions were first built using geometrical method; barycentric averaging method for the recovery of har-monic and biharmonic functions by the example of Sophie Germain equation, which models arbitrary shape elastic plate bending strains, what allows to appreciably decrease the amount of computing was first developed; the geometrical method of elastic plate bending strains modeling using the triangular element Morley, which in contrast to traditional approach doesn't use variation concepts was first developed; the interpolation polynomial fitting by means of bilinear interpolation for the built models with the square calculating template form functions properties improvement and wave making procedure was improved; the geometrical modelling method for the two- an three-dimensional serendipity family elements was evolved; the method of geometrical modeling of two-dimensional serendipity family final elements was propagated on the polar coordinates elements; the barycentric averaging method for the harmonic functions recovery using the templates, what allow superconvergence nodes, in particular for the prismatic bars torsion problem, was evolved; efficiency - branch; application - machine-building.