Abstract:
A method for the search for numerical solutions to the Dirichlet boundary value problems for nonlinear partial differential equations of the elliptic type and of an arbitrary dimension is proposed. It ensures low consumptions of memory and computer time for the problems with smooth solutions. The method is based on the modified interpolation polynomials with the Chebyshev nodes for approximation of the sought for function and on the new approach to constructing and solving the problems of linear algebra corresponding to the given differential equations. The analysis of spectra and condition numbers of matrices of the designed algorithm is made by applying the interval methods. The theorems on approximation and stability of the algorithm proposed for the linear case are proved. It is shown that the algorithm ensures an essential decrease in computational costs as compared to the classical collocation methods and to finite difference schemes.
Key words:
Dirichlet boundary value problem, decrease of computational costs, pseudospectral method, collocation method, relaxation method.
Citation:
B. V. Semisalov, “On an approach to numerical solutions of the Dirichlet problem of an arbitrary dimension”, Sib. Zh. Vychisl. Mat., 25:1 (2022), 77–95
\Bibitem{Sem22}
\by B.~V.~Semisalov
\paper On an approach to numerical solutions of the Dirichlet problem of an arbitrary dimension
\jour Sib. Zh. Vychisl. Mat.
\yr 2022
\vol 25
\issue 1
\pages 77--95
\mathnet{http://mi.mathnet.ru/sjvm798}
\crossref{https://doi.org/10.15372/SJNM20220106}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4461416}
Linking options:
https://www.mathnet.ru/eng/sjvm798
https://www.mathnet.ru/eng/sjvm/v25/i1/p77
This publication is cited in the following 1 articles:
B. V. Semisalov, “Primenenie drobno-ratsionalnykh interpolyatsii dlya resheniya kraevykh zadach s osobennostyami”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 15:4 (2022), 5–19
Citation in format AMSBIB
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