Research object - physical fields, that originate in technical systems and objects. Research target is the elaboration of the geometrical models and efficacious methods of constructive renewal of the physical fields, that originate in technical systems and objects for increasing of the estimation accuracy and decreasing of resources expense, and also investigation of interpolational properties of the discrete models for improving their characteristics. Research methods: methods of the geometrical and probabilistically-geometrical modeling for the alternative models construction over finite elements; methods of computer and physical modeling for the approbation method of the physical fields renewal, offered in the work; methods of the mathematical physics for the construction of the pagoda-function; the finite differences method, the finite element method and the Monte-Carlo method for appraising of the calculation accuracy; methods of the barycentric averaging for the calculation of the physical fields; methods of the probability theory during solving generalized Buffon task about needle. Theoretical and practical results - construction of the alternative models over different configuration finite elements, that allow to increase of the estimation accuracy with decreasing of resources expense; elaboration of the gridless method for solving the Dirichlet task in areas of complicated unprotuberant form, that allow to adequately renewal of the physical fields, that arise in technical systems and objects; discovery of the physical fields firmness relatively to the serendipian finite elements basises phenomenon, that enrich knowledge about this elements properties and give opportunity to solve the overlaying elements with different basises problem. Novelty - firstly finite elements method for solving task about rectangular cross-section prismatic beam torsion, using alternative models of serendipian family elements, is realized, that gave the opportunity to optimize the suggested models; firstly unitary models on the octagon were constructed, that include polynomial models, that are harmonic by the Laplace differential criterion and Prevalove and Kebe integral criterions, and fractionally-rational models, that are harmonic by the integral criterions, that adequately modeling the physical field with the octagonal carrier; firstly the phenomenon of the physical fields firmness relatively to the basis on the serendipian elements in the two-dimensional and three-dimensional spaces is discovered and argued analytically, that has both theoretical and practical importance during overlaying elements with different basises, for instance; firstly the efficacious gridless method for the renewal of the physical fields in areas of complicated unprotuberant form is elaborated, according to this method the mathematical models considering the particulars of the technical tasks are built and the comparing of the obtained results with the results, obtained by the help of known calculation methods and computer and physical experiments is conducted; models, that are using in the finite element method by the applying of geometrical modeling are improved, in consequence of this the calculation resources for it's realization were decreased; solving the problem of disclosing "hidden" parameters of interpolation polynomials on the two-dimensional and three-dimensional higher order serendipian elements without inputting additional nodes took a further development, consequently new classes of models were discovered and their testing with the purpose of optimization was conducted; the geometrical modelling of the hexagonal elements took a further development, that allowed to build a new unitary hexagonal model, that is harmonic by the Kebe and Prevalove integral criterions, that was successfully tested in renewal of the stationary fields tasks. Research results introduced in the Karpenko physico-mechanical institute of the national academy of sciences of Ukraine (Lviv) during appraising of temperature fields and in Ivano-Frankivsk national technical university of oil and gas, that is confirmed by the acts of introduction. Science and practice results of the dissertation may be used by constructor and project organizations for making the physical field express-analyse, in educational process during the preparation specialists in mathematical models and calculation methods branch; in science institutions and organizations, that are investigating in the direction of elaboration the mathematical models and methods of renewal of the physical fields.
