一类非线性四阶波动方程的保能混合有限元法

IF 2.6 2区 数学 Q1 MATHEMATICS, APPLIED
Mingyan He , Fengchi Tu , Jia Tian , Pengtao Sun
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引用次数: 0

摘要

本文利用Raviart-Thomas混合单元和Crank-Nicolson时间离散,提出了一类广义非线性四阶波动方程的保能混合有限元方法(FEM),该方法除得到主变量外,还同时得到了其数值梯度和拉普拉斯算子。并在理论分析中严格证明了其能量范数的节能性和最优收敛率。此外,介绍了一种特殊的投影技术,并以耦合和混合有限元形式进行了分析,以帮助证明最优收敛性。数值实验验证了所有理论结果。所开发的数值方法为解决具有非线性四阶波动方程形式的梁和薄板的物理振动问题提供了指导,这些问题在保证能量守恒的情况下长期稳定、准确。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
An energy-preserving mixed finite element method for a class of nonlinear fourth-order wave equations
In this paper, a type of energy-preserving mixed finite element method (FEM) is proposed for a class of generalized nonlinear fourth-order wave equations by means of the Raviart–Thomas mixed element and Crank–Nicolson temporal discretization in its full discretization, where besides the primary variable, its numerical gradient and Laplacian are also obtained, simultaneously, with the energy conservation property and optimal convergence rates in their energy norms that are rigorously proved in the theoretical analyses. In addition, a particular projection technique is introduced and analyzed in a coupling and mixed finite element form to assist in proving the optimal convergence properties. Numerical experiments are carried out to validate all theoretical results. The developed numerical method provides a guide to solve physical vibration problems of beams and thin plates, which own the form of nonlinear fourth-order wave equations, stably and accurately for a long time in a way that ensures energy conservation.
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来源期刊
CiteScore
5.40
自引率
4.20%
发文量
437
审稿时长
3.0 months
期刊介绍: The Journal of Computational and Applied Mathematics publishes original papers of high scientific value in all areas of computational and applied mathematics. The main interest of the Journal is in papers that describe and analyze new computational techniques for solving scientific or engineering problems. Also the improved analysis, including the effectiveness and applicability, of existing methods and algorithms is of importance. The computational efficiency (e.g. the convergence, stability, accuracy, ...) should be proved and illustrated by nontrivial numerical examples. Papers describing only variants of existing methods, without adding significant new computational properties are not of interest. The audience consists of: applied mathematicians, numerical analysts, computational scientists and engineers.
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