title: Ð ÐµÑÐµÐ½Ð¸Ðµ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÑÐ°ÑÑÐ¾Ð½Ð° Ñ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸ÐµÐ¼ Ð°Ð´Ð°Ð¿ÑÐ¸Ð²Ð½ÑÑ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð¾Ð² Ð¿ÐµÑÐµÑÑÑÐ¾ÐµÐ½Ð¸Ñ ÑÐµÑÐ¾Ðº
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: Solving the Poisson equation using adaptive mesh refinement algorithms
Creators: name: ÐÐ²Ð°Ð½Ð¾Ð²Ð°, ÐÐ½Ð½Ð°
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: 2024
Subjects: ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ ÐÑÐ°ÑÑÐ¾Ð½Ð°
ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÐµ Ð¼ÐµÑÐ¾Ð´Ñ
Ð¼ÐµÑÐ¾Ð´ ÐºÐ¾Ð½ÐµÑÐ½ÑÑ ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ¾Ð²
Ð°Ð´Ð°Ð¿ÑÐ¸Ð²Ð½ÑÐµ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ñ
Poisson equation
numerical methods
finite element method
adaptive algorithms

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description: This work is devoted to solving the Dirichlet problem for the Poisson equation in 2D by the finite element method and refining the solution using adaptive algorithms based on the use of a posteriori estimates of the error of the solution. In the course of the work, the following problems were posed and solved: implementation of the finite element method for the Poisson equation with the Dirichlet boundary condition; implementation of the mesh adaptation algorithm; solving several problems and finding a local error distribution using data on the exact solution of the problem and a posteriori methods for indicating the error of the solution; adaptation of the mesh based on the values of the error indicators; comparative analysis of the effectiveness of adaptive and classical algorithms, as well as error indicators. In the course of a comparative analysis of the classical and adaptive approaches to clarifying the solution, using the example of several tasks, it was shown that the adaptive approach is more efficient, it allows you to take into account the features of the exact solution and save computational resources. It was demonstrated how the features of the exact solution affect the error distribution. It has been shown that the gradient averaging method and the implicit residual method make it possible to estimate the error distribution in trivial cases.