Astionenko I. Models of Functions Approximation by Multiparameter Polynomials of Serendipity Family

Purpose of the research is the development of a combined algebro-geometric method of modelling of the interpolation polynomials for the functions of two arguments, construction of the new multiparameter models of serendipity finite elements. Object of the dissertation is the finite elements of serendipity family. Subject of the research is the ways of interpolation of the functions of two arguments on the elements of serendipity family with the aim of creating the alternative models. The following methods were used in the research: method of analytical geometry; numerical methods; method of probability theory; method of cognitive computer graphics; method of mathematical physics, method of variational calculus, method of finite element. For the first time: the inverse problem of serendipity approximation, which led to the creation of combined (algebraic and geometric) method of building the serendipity finite elements basises was formulated; new multiparameter models of the finite elements of the higher orders on the surface with governing parameter are built; the theorems on infinite aggregate of multiparameter serendipity models of the higher orders and the uniqueness of standard models are formulated and proved constructively. The proposed alternative models of serendipity approximations can be used in the packages of application programs which are geared to finite-element calculations. This can be applied in the problems of thermophysics, geomechanics, movement of liquid and gas, structural mechanics, problems of elasticity and plasticity theory.