title: ÐÐ´Ð°Ð¿ÑÐ°ÑÐ¸Ñ ÑÐµÑÐ¾Ðº Ð¿ÑÐ¸ ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ð¼ ÑÐµÑÐµÐ½Ð¸Ð¸ ÐºÐ»Ð°ÑÑÐ¸ÑÐµÑÐºÐ¸Ñ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¹ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð¹ ÑÐ¸Ð·Ð¸ÐºÐ¸
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: Adaptation of grids in the numerical solution of classical equations of mathematical physics
Creators: name: Ð¨Ð°Ð¾ Ð¦Ð·ÑÑÐ¸
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¸Ð²Ð½Ð°Ñ ÑÐµÑÐºÐ°
Ð°Ð¿Ð¾ÑÑÐµÑÐ¸Ð¾ÑÐ½Ð°Ñ Ð¾ÑÐµÐ½ÐºÐ°
numerical methods
finite element method
adaptive mesh
posterior estimate

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description: The topic of the final qualification work: Adaptation of meshes in the numerical solution of classical equations of mathematical physics. This work is devoted to the consideration of the finite element method for solving a one-dimensional boundary value problem of mathematical physics, comparing errors on a uniform mesh and on an adaptive mesh, and constructing an indicator for analyzing the behavior of the resulting solution without using an analytical solution. Tasks that were solved during the work: 1. Implementation of the finite element method on a uniform mesh. 2. Implementation of the finite element method on an adaptive mesh. 3. Constructing an indicator by averaging method to analyze the behavior of the resulting solution. Numerical experiments have shown that in the finite element method, the adaptation of the mesh allows increasing the accuracy of the solution compared to a uniform mesh with the same number of elements, and on the other hand, the adaptation also saves computational costs to achieve the same accuracy compared to a uniform mesh.The finite element method with adaptation of mesh usually consists of the following four processes: 1. Solve the problem on this mesh. 2. Build an indicator to evaluate the error. 3. Mark the elements that need to be shredded. 4. Grind the mesh according to the adaptation algorithm and get a new mesh.