Geodesic
A conservative finite element scheme for the Kirchhoff equation
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 165 (2023) no. 2, pp. 115-131

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This article presents an implicit two-layer finite element scheme for solving the Kirchhoff equation, a nonlinear nonlocal equation of hyperbolic type with the Dirichlet integral. The discrete scheme was designed considering the solution of the problem and its derivative for the time variable. It ensures total energy conservation at a discrete level. The use of the Newton method was proven to be effective for solving the scheme on the time layer despite the nonlocality of the equation. The test problems with smooth solutions showed that the scheme can define both the solution of the problem and its time derivative with an error of $O(h^2+\tau^2)$ in the root-mean-square norm, where $\tau$ and $h$ are the grid steps in time and space, respectively.
Keywords: Kirchhoff equation, finite element method, Petrov–Galerkin method, Newton method.
Mots-clés : implicit scheme 
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     title = {A conservative finite element scheme for the {Kirchhoff} equation},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
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R. Z. Dautov; M. V. Ivanova. A conservative finite element scheme for the Kirchhoff equation. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 165 (2023) no. 2, pp. 115-131. http://geodesic.mathdoc.fr/item/UZKU_2023_165_2_a1/