Geodesic
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, mehanika, kompʹûternye nauki, Tome 31 (2021) no. 1, pp. 3-18

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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
Mots-clés : convergence estimation. 
@article{VUU_2021_31_1_a0,
     author = {A. S. Il'inskii and I. S. Polyanskii and D. E. Stepanov},
     title = {On the convergence of the barycentric method in solving internal {Dirichlet} and {Neumann} problems in $ \mathbb{R}^2$ for the {Helmholtz} equation},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {3--18},
     publisher = {mathdoc},
     volume = {31},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2021_31_1_a0/}
}
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A. S. Il'inskii; I. S. Polyanskii; D. E. Stepanov. 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, mehanika, kompʹûternye nauki, Tome 31 (2021) no. 1, pp. 3-18. http://geodesic.mathdoc.fr/item/VUU_2021_31_1_a0/