DOI: https://doi.org/10.26089/NumMet.v17r446

A fast nonlocal algorithm for solving Neumann-Dirichlet boundary value problems with error control

Authors

  • B.V. Semisalov

Keywords:

boundary value problem
fast algorithm
estimation of error
collocation method
relaxation method
nonlocal algorithm without saturation

Abstract

A method for searching numerical solutions to Neumann-Dirichlet boundary value problems for differential equations of elliptic type is proposed. This method allows reaching a desired accuracy with low consumption of memory and computer time. The method adapts the properties of best polynomial approximations for construction of algorithms without saturation on the basis of nonlocal Chebyshev approximations. A new approach to the approximation of differential operators and to solving the resulting problems of linear algebra is also proposed. Estimates of numerical errors are given. A high convergence rate of the proposed method is substantiated theoretically and is shown numerically in the case of problems with Cr-smooth and C∞-smooth solutions. Expressions for arrays approximating the differential operators in problems with various types of boundary conditions are obtained. These expressions allow the reader to quickly implement the method «from scratch».

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Published

2016-11-26

Issue

Vol. 17 (2016): Issue 4.

Section

Section 1. Numerical methods and applications

Author Biography

B.V. Semisalov

Novosibirsk State University,
Department of Mathematics and Mechanics
• Assistant

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