A Conservative Finite Element Scheme for the Kirchhoff Equation

R. Dautov, M. V. Ivanova
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Abstract

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(h2+τ2) in the root-mean-square norm, where τ and h are the grid steps in time and space, respectively.
基尔霍夫方程的保守有限元方案
本文提出了一种隐式双层有限元方案,用于求解基尔霍夫方程(一种具有迪里希特积分的双曲型非线性非局部方程)。离散方案的设计考虑了问题的求解及其对时间变量的导数。它确保了离散水平上的总能量守恒。尽管方程具有非局部性,但使用牛顿法在时间层上求解该方案被证明是有效的。具有平滑解的测试问题表明,该方案可以定义问题的解及其时间导数,均方根规范误差为 O(h2+τ2),其中 τ 和 h 分别为时间和空间的网格步长。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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