{"title":"On an Approach to the Numerical Solution of Dirichlet Problems of Arbitrary Dimensions","authors":"B. V. Semisalov","doi":"10.1134/s1995423922010062","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A method of the numerical solution of Dirichlet boundary value problems for nonlinear partial differential equations of the elliptic type and of arbitrary dimensions is proposed. It takes little memory and computer time for problems with smooth solutions. The method is based on modified interpolation polynomials with Chebyshev nodes to approximate the sought-for function and on a new approach to constructing and solving the linear algebraic problems corresponding to the initial differential equations. The spectra and condition numbers of the matrices formed by the algorithm are analyzed by using interval methods. Theorems on approximation and stability of the algorithm are proved in the linear case. It is shown that the algorithm provides a considerable decrease in computational costs as compared to the classical collocation and finite difference methods.</p>","PeriodicalId":43697,"journal":{"name":"Numerical Analysis and Applications","volume":"64 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerical Analysis and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1995423922010062","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 2
Abstract
A method of the numerical solution of Dirichlet boundary value problems for nonlinear partial differential equations of the elliptic type and of arbitrary dimensions is proposed. It takes little memory and computer time for problems with smooth solutions. The method is based on modified interpolation polynomials with Chebyshev nodes to approximate the sought-for function and on a new approach to constructing and solving the linear algebraic problems corresponding to the initial differential equations. The spectra and condition numbers of the matrices formed by the algorithm are analyzed by using interval methods. Theorems on approximation and stability of the algorithm are proved in the linear case. It is shown that the algorithm provides a considerable decrease in computational costs as compared to the classical collocation and finite difference methods.
期刊介绍：
Numerical Analysis and Applications is the translation of Russian periodical Sibirskii Zhurnal Vychislitel’noi Matematiki (Siberian Journal of Numerical Mathematics) published by the Siberian Branch of the Russian Academy of Sciences Publishing House since 1998.
The aim of this journal is to demonstrate, in concentrated form, to the Russian and International Mathematical Community the latest and most important investigations of Siberian numerical mathematicians in various scientific and engineering fields.
The journal deals with the following topics: Theory and practice of computational methods, mathematical physics, and other applied fields; Mathematical models of elasticity theory, hydrodynamics, gas dynamics, and geophysics; Parallelizing of algorithms; Models and methods of bioinformatics.
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