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Russia Moscow; Orel
Year
2021
Volume
31
Issue
1
Pages
3-18
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Section Mathematics
Title On the convergence of the barycentric method in solving internal Dirichlet and Neumann problems in $\mathbb{R}^2$ for the Helmholtz equation
Author(-s) Il’inskii A.S.a, Polyanskii I.S.b, Stepanov D.E.b
Affiliations Lomonosov Moscow State Universitya, The Academy of Federal Security Guard Service of the Russian Federationb
Abstract The application of the barycentric method for the numerical solution of Dirichlet and Neumann problems for the Helmholtz equation in the bounded simply connected domain $\Omega\subset\mathbb{R}^2$ is considered. The main assumption in the solution is to set the $\Omega$ boundary in a piecewise linear representation. A distinctive feature of the barycentric method is the order of formation of a global system of vector basis functions for $\Omega$ via barycentric coordinates. The existence and uniqueness of the solution of Dirichlet and Neumann problems for the Helmholtz equation by the barycentric method are established and the convergence rate estimate is determined. The features of the algorithmic implementation of the method are clarified.
Keywords internal Dirichlet and Neumann problems, Helmholtz equation, arbitrary polygon, barycentric method, Galerkin method, barycentric coordinates, convergence estimation
UDC 519.632
MSC 35J05, 65N12
DOI 10.35634/vm210101
Received 24 June 2020
Language Russian
Citation Il’inskii A.S., Polyanskii I.S., Stepanov D.E. On the convergence of the barycentric method in solving internal Dirichlet and Neumann problems in $\mathbb{R}^2$ for the Helmholtz equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 1, pp. 3-18.
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