Unsaturated Algorithms for the Numerical Solution of Elliptic Boundary Value Problems in Smooth Axisymmetric Domains Full article
| Journal |
Siberian Advances in Mathematics
ISSN: 1055-1344 , E-ISSN: 1934-8126 |
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| Output data | Year: 2022, Volume: 32, Number: 1, Pages: 157-185 Pages count : 29 DOI: 10.1134/S1055134422030014 | ||
| Tags | Laplace equation, axisymmetric domain, unsaturated numerical method, wellconditioned problem, exponential convergence. | ||
| Authors |
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| Affiliations |
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Funding (1)
| 1 | Sobolev Institute of Mathematics | FWNF-2022-0004 |
Abstract:
A fundamentally new, unsaturated, method is constructed for the numerical solution
of the Laplace equation in a smooth axisymmetric domain of rather general shape. An essential
feature of this method is lack of the leading error term O(m−r), where r is a fixed integer with
r > 2. As a result, the method automatically adjusts to the excess (extraordinary) smoothness of
solutions of problems. The method provides us with a new computational tool whose discretization
inherits both differential and spectral characteristics of the operator of the elliptic problem under
consideration. This allows us to take efficiently into account the fact that the domain is axisymmetric
which is a stumbling block for numerical methods with the leading error term. Our result is
of a fundamental interest because, for C ∞-smooth solutions, computer constructs a numerical
solution (up to a slowly growing multiplier) with an absolutely sharp exponential error estimate.
The sharpness is caused by the asymptotics of the Aleksandrovm-width of the compact set of C ∞-
smooth functions that contains the exact solution of the problem. This asymptotics is presented by
a function that exponentially decays as the integer parameter m grows.
Cite:
Belykh V.N.
Unsaturated Algorithms for the Numerical Solution of Elliptic Boundary Value Problems in Smooth Axisymmetric Domains
Siberian Advances in Mathematics. 2022. V.32. N1. P.157-185. DOI: 10.1134/S1055134422030014 Scopus РИНЦ OpenAlex
Unsaturated Algorithms for the Numerical Solution of Elliptic Boundary Value Problems in Smooth Axisymmetric Domains
Siberian Advances in Mathematics. 2022. V.32. N1. P.157-185. DOI: 10.1134/S1055134422030014 Scopus РИНЦ OpenAlex
Original:
Белых В.Н.
Ненасыщаемые алгоритмы численного решения эллиптических краевых задач в гладких осесимметричных областях
Математические труды. 2022. Т.25. №1. С.3-50. DOI: 10.33048/mattrudy.2022.25.101 РИНЦ
Ненасыщаемые алгоритмы численного решения эллиптических краевых задач в гладких осесимметричных областях
Математические труды. 2022. Т.25. №1. С.3-50. DOI: 10.33048/mattrudy.2022.25.101 РИНЦ
Dates:
| Submitted: | Jan 17, 2022 |
| Published online: | May 12, 2022 |
Identifiers:
| Scopus: | 2-s2.0-85137755186 |
| Elibrary: | 51489522 |
| OpenAlex: | W4294335368 |