USING THE METHOD OF WEIGHTED RESIDUALS IN THE FEM FORM FOR THE APPROXIMATION OF TWO-DIMENSIONAL DISTRIBUTIONS

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The paper is devoted to the description of the method of two-dimensional approximation of data using algorithms of the method of weighted residuals with piecewise specified basis functions in the implementation of the Finite Elements Method. The mathematical formulation of the problem and the main relations of the method for two-dimensional regions are provided. A triangular finite element is involved in the simulation. Information is given about the program developed according to the algorithms of this method, the result of which is the selection of regions of a two-dimensional figure with the specified values of the analyzed parameters. By use of randomly distributed coordinates, the verification of the results for different data sets was performed and a satisfactory correspondence of the approximated and input data was obtained. Additional software, including programs written in Python and a web application created using the Laravel framework in PHP, has been developed to create a database that is scheduled to be processed by approximation. The PyQt5, OSMnx, requests, aiohttp, BeautifulSoup libraries and the PostgreSQL database management system were used to implement the project tasks. Algorithms of the developed software are described. An example of the reproduction of the locations of cities on the map of Ukraine is provided. The developed program of two-dimensional approximation was applied to the analysis of spatio-temporal dynamics of the emergence of cities on the territory of Ukraine. As an example, data on the founding of cities in the period from 800 to 1200 AD were considered and the main urbanized regions at that time were determined. The results of this Finite Element approximation were obtained for the purpose of preparing data for further modeling of sociodynamic processes in the terrain of Ukraine. The obtained results in the form of coordinates of the borders of regions that had a high level of urbanization can be used in various scientific studies, primarily in historical sciences.