title: Ð§Ð¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ Ð·Ð°Ð´Ð°Ñ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð¹ ÑÐ¸Ð·Ð¸ÐºÐ¸ Ð² 2D Ñ Ð¸Ð½Ð´Ð¸ÐºÐ°ÑÐ¸ÐµÐ¹ Ð¿Ð¾Ð³ÑÐµÑÐ½Ð¾ÑÑÐ¸ ÑÐµÑÐµÐ½Ð¸Ð¹
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Numerical solution of mathematical physics problems in 2D with indication of the error of solutions
Creators: name: Ð¥Ð°ÑÐ¸ÑÐ¾Ð²Ð°, Ð¢Ð°Ð¸ÑÐ¸Ñ
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2022
Subjects: ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÐµ Ð¼ÐµÑÐ¾Ð´Ñ
Ð¼ÐµÑÐ¾Ð´ ÐºÐ¾Ð½ÐµÑÐ½ÑÑ ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ¾Ð²
Ð°Ð´Ð°Ð¿ÑÐ°ÑÐ¸Ñ ÑÐµÑÐ¾Ðº
Ð¸Ð½Ð´Ð¸ÐºÐ°ÑÐ¾ÑÑ
numerical methods
finite element method
mesh adaptation
indicators

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description: The subject of the graduate qualification work is Â«Numerical solution of mathematical physics problems in 2D with indication of the error of solutionsÂ». The given work is devoted to the consideration of the finite element method for the numerical solution of two-dimensional problems of mathematical physics, and the analysis of the results of using error indicators of the solution using adaptive meshes. The research set the following goals:Implementation of the finite element method and classical indication methods. Solving the problem and indicating the local distribution of the error of the solution over the area using the implemented methods. Adaptation of the mesh using different indication methods. Comparison of solutions obtained on uniform and adaptive meshes using different display methods.The finite element method and two indication methods were implemented: an indicator based on gradient averaging, and an indicator based on an explicit residual method. Numerical experiments were carried out, during which the implemented methods were used to solve boundary value problems of mathematical physics. As a result, the work of the indicators was demonstrated, as well as their comparative analysis was carried out. The indicator based on gradient averaging turned out to be more versatile and accurate than the indicator based on the explicit residual method. An adaptive algorithm is implemented and its comparison with uniform meshing of the domain is carried out. The use of an adaptive algorithm turned out to be more efficient in terms of the number of nodes (D.O.F.s) required to achieve the desired accuracy.